# The infall region as a complementary probe to cluster abundance

C. T. Mpetha,1,2,3 J. E. Taylor,2,3 Y. Amoura,2,3 R. Haggar2,3
1Institute for Astronomy, School of Physics and Astronomy, University of
Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, United
Kingdom
2Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, Ontario
N2L 3G1, Canada
3 Department of Physics and Astronomy, University of Waterloo, 200 University
Avenue West, Waterloo, Ontario N2L 3G1, Canada
E-mail<EMAIL_ADDRESS>(CTM)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Galaxy cluster abundance measurements provide a classic test of cosmology.
They are most sensitive to the evolved amplitude of fluctuations, usually
expressed as $S_{8}=\sigma_{8}\sqrt{\Omega_{m}/0.3}$. Thus, abundance
constraints exhibit a strong degeneracy between $\sigma_{8}$ and $\Omega_{\rm
m}$, as do other similar low-redshift tests such as cosmic shear. The mass
distribution in the infall region around galaxy clusters, where material is
being accreted from the surrounding field, also exhibits a cosmological
dependence, but in this case it is nearly orthogonal to the $S_{8}$ direction
in the $\Omega_{m}$–$\sigma_{8}$ plane, making it highly complementary to halo
abundance or cosmic shear studies. We explore how weak lensing measurements of
the infall region might be used to complement abundance studies, considering
three different tests. The splashback radius is a prominent feature of the
infall region; we show that detection of this feature in lensing data from the
Euclid survey could independently constrain $\Omega_{\rm m}$ and $\sigma_{8}$
to $\pm 0.05$. Another feature, the depletion radius where the bias reaches a
minimum, also shows cosmological dependence, though it is challenging to
observe in practice. The strongest constraints come from direct measurements
of the shear profile in the infall region at $2$–$4\,r_{200{\rm c}}$.
Combining the latter with abundance constraints such as those reported from
SRG$/$eROSITA should reduce the area of the error contours by an estimated
factor of $1.2$ using a sample of clusters observed by the UNIONS survey, or a
factor of $3$ using clusters observed by the Euclid Wide survey over a broader
range of redshift.

###### keywords:

gravitational lensing: weak – methods: observational – galaxies: clusters:
general – galaxies: groups: general – galaxies: haloes – cosmological
parameters

††pubyear: 2024††pagerange: The infall region as a complementary probe to
cluster abundance–B

##  1 Introduction

Galaxy clusters provide an excellent test-bed for astrophysics and cosmology.
Their overall abundance depends sensitively on the evolved amplitude of
fluctuations, often expressed as $S_{8}=\sigma_{8}\sqrt{\Omega_{m}/0.3}$.
Because $S_{8}$ combines the effects of early power, characterised by the
linearly extrapolated amplitude of fluctuations on 8 $h^{-1}$ Mpc scales,
$\sigma_{8}$, and late-time growth, characterised by the matter density
parameter $\Omega_{m}$, abundance measurements have a strong degeneracy in the
$\Omega_{m}$–$\sigma_{8}$ plane (Viana & Liddle, 1996; Bartlett, 1997), as do
other, similar low-redshift probes of cosmological structure, such as cosmic
shear (Kilbinger, 2015). The assembly rate of clusters and their underlying
dark matter halos has a different dependence on $\Omega_{\rm m}$ and
$\sigma_{8}$ (Amoura et al., 2021), and could provide an alternative
cosmological test if it could be quantified through observations of cluster
structure, an idea first proposed in the early days of cluster cosmology
(Richstone et al., 1992; Evrard et al., 1993; Mohr et al., 1995).

A number of previous studies have investigated the link between cluster
structure, assembly history, and cosmology. The density profile of a dark
matter halo within its virialised region has been investigated as a
cosmological probe, via measurements of concentration (Dolag et al., 2004) or
halo sparsity (Balmès et al., 2014). Predictions in the inner parts of
observed clusters remain uncertain, however, because of poorly constrained
baryonic effects (e.g. Debackere et al., 2021). Furthermore, the constrains on
$\Omega_{\rm m}$ and $\sigma_{8}$ from these methods have a similar degeneracy
to abundance measurements (Ettori et al., 2011; Corasaniti et al., 2021), and
so combining either of them with abundance constraints may not lead to
significant improvement. In contrast, the infall region outside the virial
radius, where matter is currently being accreted onto the halo, can reveal
information about its recent growth history (Shi, 2016). The current growth
rate has in turn been shown to be a useful cosmological probe (Amoura et al.,
2023), with a degeneracy direction almost orthogonal to $S_{8}$ over a range
of mass and redshift. Its independent measurement, in the same data used for
abundance and cosmic shear studies, could break the $\Omega_{\rm
m}$–$\sigma_{8}$ degeneracy, significantly improving the precision of
abundance constraints.

The infall region includes a number of distinct features. The splashback
radius $r_{\rm sp}$, the apocenter of the first orbit of material after it has
been accreted into a dark matter halo (Diemer & Kravtsov, 2014; Adhikari et
al., 2014; More et al., 2015), is the most widely studied feature and is
sometimes proposed as a more physical boundary than the virial radius, as it
mitigates the impact of pseudo-evolution (Diemer et al., 2013; Diemer, 2020).
The so-called ‘depletion zone’ (Fong & Han, 2021) is the region around a halo
from which material has been accreted, and hence is depleted with respect to
the expected background. The ‘depletion radius’, the radius where the bias
(defined below) reaches a minimum, is another possible definition for the halo
boundary (Zhou & Han, 2023). Another feature of the depletion zone, the ‘inner
depletion radius’, has been shown to match up well with the “optimal halo
exclusion radius" defined in García et al. (2021). Both the splashback radius
and depletion radius are sensitive to the mass accretion history of a halo
(Shin & Diemer, 2023; Gao et al., 2023). There is then an interplay between
the formation time, the recent accretion history, and these features of the
density profile, from which we may be able to derive cosmological information.

The splashback feature has been detected significantly using galaxy number
density profiles (e.g. Chang et al., 2018; Zürcher & More, 2019; Murata et
al., 2020; Bianconi et al., 2021; Shin et al., 2021; Adhikari et al., 2021;
Rana et al., 2023) and weak lensing (e.g. Chang et al., 2018; Contigiani et
al., 2019; Shin et al., 2021; Contigiani et al., 2023), which has also been
used to observe the depletion zone (Fong et al., 2022). Measuring the
splashback radius through weak lensing is advantageous as it does not require
knowledge of the galaxy bias, it allows the stacking of many dark matter
halos, and provides mass-calibration for the observed sample. As this feature
can be measured with good precision, using it as a cosmological test seems
feasible. (Diemer et al., 2017) made an initial study of the variation of the
splashback radius with cosmology, testing WMAP and Planck cosmologies, and
models with scale-free power spectra. Haggar et al. (2024), henceforth H24,
recently made a more detailed study of the cosmological dependence, showing
that on cluster scales, the variation of $r_{\rm sp}$ is close to orthogonal
to the $S_{8}$ direction in the $\Omega_{\rm m}$–$\sigma_{8}$ plane. There
also has been work exploring how measurements of the splashback radius can be
used to constrain alternative gravity models (Adhikari et al., 2018; Pizzardo
et al., 2024). To our knowledge there has been no systematic investigation of
how the depletion radius, or the general form of the density profile in the
infall zone, depends on cosmology.

In this work, we use dark-matter-only simulations to determine how features in
the infall region of cluster-mass halos vary with cosmology. We then
investigate how accurately these features could be measured in present and
forthcoming weak lensing surveys. Our goal is to determine the best feature(s)
to measure, and also the best cluster mass and redshift range to use for
cosmological tests. To illustrate the complementarity with abundance studies,
we will focus on the recent constraints derived from clusters detected in the
SRG$/$eROSITA All-Sky Survey (Ghirardini et al., 2024). Beyond these results,
there are exciting prospects for further improvements in abundance
measurements from several present and forthcoming surveys, including the
UNIONS (Chambers et al., 2020), Euclid-Wide (Sartoris et al., 2016), and
surveys by the Nancy Grace Roman Space Telescope (Dore et al., 2019) and the
Vera C. Rubin Observatory (Ivezić et al., 2019).

The outline of the paper is as follows. In Section 2, we describe the
cosmological simulations used in this work. Section 3 discusses the mean
density profile of the infall region, its main features, and their
cosmological dependence. We also describe our profile fitting process in
detail. Section 4 explains how the 3D profiles are projected to produce the 2D
mass profiles measured by lensing, and how these projected profiles are then
fitted. Section 5 presents forecasts for the constraining power of the infall
region, and in Section 6 we consider some anticipated challenges in applying
this test to observations. We conclude in Section 7.

##  2 Simulations

We use a suite of 21 dark-matter-only simulations described in Amoura (2023)
(see also Amoura et al. 2024, in preparation). The simulated cosmologies range
over the combinations of $\Omega_{\rm m}$ and $\sigma_{8}$ shown in Table 1.
Other cosmological parameters are fixed, with a Hubble parameter
$H_{0}=100\,h=70\,$km s${}^{-1}\,$Mpc-1, a baryon density $\Omega_{\rm
b}=0.0482$, and a spectral tilt $n_{s}=0.965$.

Each simulation was performed using Gadget 4 (Springel et al., 2021) with
$1024^{3}$ particles in a $500\,$Mpc$/h$ box, giving a particle mass
$\Omega_{\rm m}\times(3.23\times 10^{10}M_{\odot})$, and a softening length of
$2.5\,$kpc. Dark matter halos were identified using the Amiga Halo Finder
(AHF) (Knollmann & Knebe, 2009), down to a lower mass limit of $\sim
10^{12}M_{\odot}\,h^{-1}$, and their evolution was traced from $z\\!\sim\\!30$
to $z=0$ in $119$ snapshots (except for two initial runs denoted by an $o$ in
Table 1, which only had output saved at $z=0$).

In what follows, comoving distances are written $c\,$Mpc$\,h^{-1}$. Halo
masses $M_{200{\rm c}}$ and radii $r_{200{\rm c}}$ are defined as the mass
enclosed within, and radius of, a spherical region 200 times the critical
density $\rho_{c}$, respectively.

As discussed in H24, for cluster-mass halos, baryonic effects are more or less
negligible in the infall region, relative to the cosmological signal we
consider. This conclusion is also supported by other works comparing dark-
matter-only simulations with hydrodynamical simulations including baryons
(e.g. Haggar et al., 2021; O’Neil et al., 2021; Towler et al., 2024). Thus, we
have ignored baryonic effects in our analysis. These would be more significant
on smaller mass scales, as discussed below, limiting the accuracy of the
method on these scales.

Table 1: Combinations of $\Omega_{\rm m}$ and $\sigma_{8}$ assumed in the cosmological simulations of Amoura (2023) $\sigma_{8}$ $\Omega_{\rm m}$ | 0.2 | 0.25 | 0.3 | 0.35 | 0.4
---|---|---|---|---|---
1.0 | x |  |  |  |
0.95 |  |  |  |  |
0.9 | x | x | x | x | x
0.85 |  | x | x | x |
0.8 | $o$ | x | x | x | x
0.75 |  | x | x | x |
0.7 | x |  | x | $o$ | x

##  3 Features of the Infall Region

###  3.1 Calculating the mean density profile

Given a halo catalogue for each simulation and output redshift, we determine
the mean density profile in a given mass range as follows. The density profile
of each individual halo is found by counting the number of particles in 400
logarithmically spaced shells from $r=0.01\,$Mpc$/h$ to $r=20\,$Mpc$\,h^{-1}$,
and dividing their total mass by the volume of the shell. Shells are centred
on the most bound particle in the halo; the potential impact of mis-centering
is discussed in Section 6. To overcome the shot noise present in individual
profiles and match the quantities usually measured in lensing observations, we
average them, calculating the mean density profile for all halos in a given
mass range. Only halos with more than 200 particles are included, and any halo
flagged as a subhalo by AHF is removed from the sample. Errors on the mean
profile are computed using 500 bootstrap realisations.

###  3.2 Fitting the density profile

To fit the density profile in the infall region, we adopt the model of Diemer
(2022). This model is designed to fit to both the inner and outer regions of a
dark matter halo in a way that reflects the dynamics of the material. It
includes orbiting and infalling density terms:

$\displaystyle\rho(r)$ $\displaystyle=\rho_{\rm orbit}(r)+\rho_{\rm
infall}(r)$ (1) $\displaystyle\rho_{\rm orbit}(r)$
$\displaystyle=\rho_{s}\,\exp{\left(-\frac{2}{\alpha}\left[\left(\frac{r}{r_{s}}\right)^{\alpha}-1\right]-\frac{1}{\beta}\left[\left(\frac{r}{r_{t}}\right)^{\beta}-\left(\frac{r_{s}}{r_{t}}\right)^{\beta}\right]\right)}$
$\displaystyle\rho_{\rm infall}(r)$
$\displaystyle=A\,\left(1+\delta_{1}/\sqrt{\left(\delta_{1}/\delta_{\rm
max}\right)^{2}+\left(r/r_{\rm pivot}\right)^{2s}}\right)$ (2)

The model has $9$ free parameters
$\rho_{s},\alpha,\beta,r_{s},r_{t},A,\delta_{1}$, and $\delta_{\rm max}$, with
$r_{\rm pivot}$ being a fixed pivot scale that is set to

$r_{\rm pivot}(z)=\left(\frac{3M}{4\pi\times 200\,\rho_{\rm
m,P}(z)}\right)^{1/3}\,$ (3)

where $\rho_{\rm m,P}(z)=(1+z)^{3}\rho_{\rm m,P}(z=0)=\Omega_{\rm m}^{\rm
Planck}\rho_{c,0}(1+z)^{3}$ for a Planck 2018 cosmology (Planck Collaboration
et al., 2020), and $\Omega_{\rm m}^{\rm Planck}=0.316$. When fitting stacked
halos with a range of masses, $M$ is taken to be the minimum mass of this
range. When fitting comoving density profiles, the $(1+z)^{3}$ term is
dropped. In this model, the inner part of the profile is described by a
modified Einasto profile (Einasto, 1965) with an additional exponentially
decaying truncation term, controlled by the truncation radius $r_{t}$. For the
outer profile, $\delta_{1}$ and $\delta_{\rm max}$ are the overdensity at
$r_{\rm pivot}$ and in the centre respectively, while $s$ is the power-law
slope of the infalling term. Note that the matter density $\rho_{\rm m}$ in
the original expression (Eq. (15) in Diemer 2022) has been replaced with an
amplitude parameter $A$, as the value of $\rho_{\rm m}$ is not known a priori.

Fits are performed over the range $r=0.06-20\,$Mpc$\,h^{-1}$ when fitting the
intrinsic 3D profiles, and $r=0.1-10\,r_{\rm pivot}\,$Mpc$\,h^{-1}$ when
fitting mock observed 2D profiles, to avoid the impact of mis-centering (see
Section 6.4). The python package lmfit (Newville et al., 2023) is used as a
convenience wrapper incorporating the least-squares fitting function in scipy
(Virtanen et al., 2020) with the Trust Region Reflective method. The prior
ranges for the fitted values are given in Table 1 of Diemer (2022) and we
follow the fitting procedure described in Appendix A1 of that work, which is
designed to avoid becoming trapped in a local minima of $\chi^{2}$.

Examples of mean 3D density profiles for $z=0$ halos, and the corresponding
fit, can be seen in the left panel of Fig. 1.

Figure 1: Left: Mean 3D density profiles for halos with $M\geq
10^{14}M_{\odot}\,h^{-1}$ at $z=0$ (solid lines) for two simulations, with
$\Omega_{\rm m}=0.2$, $\sigma_{8}=1.0$ (thick magenta line) and $\Omega_{\rm
m}=0.4$, $\sigma_{8}=0.7$ (thin green line). The dotted lines show the best
fit models of Diemer (2022), which are almost indistinguishable from the data.
Also labelled are the splashback (triangle) and depletion (star) radii for
each profile. Middle: The logarithmic gradient $d\ln(\rho)/d\ln(r)$, whose
minimum defines the splashback radius $r_{\rm sp}$. Right: The bias profile
$b(r)$, whose minimum defines the depletion radius $r_{\rm cd}$.

###  3.3 Location of the splashback radius

Given our fits to the mean density profiles, we can proceed to test the
cosmological dependence of the splashback radius and other features. The
splashback radius is identified by finding the minimum of the gradient of the
logarithmic density profile, outlined in the middle panel of Fig. 1. It occurs
outside the virial radius $r_{200{\rm c}}$, and is dependent on the halo
accretion rate (Shin & Diemer, 2023), which in turn is sensitive to cosmology
(Amoura et al., 2023). H24 demonstrate that for cluster-mass halos, the
splashback radius varies in a direction nearly orthogonal to $S_{8}$ in the
$\Omega_{\rm m}$–$\sigma_{8}$ plane. The relative simplicity of identifying
$r_{\rm sp}$ makes it a promising avenue to test cosmology in the infall
region. The splashback feature is also easiest to observe in clusters, where
the drop in density is sharper compared to lower mass halos.

###  3.4 Location of the depletion radius

The depletion radius is a more complicated feature of the infall region. The
bias profile for a halo is defined by

$b(r)=\frac{\xi_{\rm hm}(r)}{\xi_{\rm
mm}(r)}=\frac{\langle\delta(r)\rangle}{\xi_{\rm mm}(r)}\,.$ (4)

and is related to its density profile though

$\rho(r)=\rho_{\rm m}\left[b(r)\,\xi_{\rm mm}(r)+1\right]\,.$ (5)

Clearly, knowledge of the matter-matter correlation function is required to
calculate the bias profile from the density profile, and this function is
itself cosmology-dependent. The impact of uncertainty in the matter-matter
correlation function on calculation of the bias profile is discussed further
in Section 6.2. The characteristic depletion radius is given by the minimum of
the bias profile, seen in the right-hand panel of Fig. 1. For the cluster-mass
halos shown in the figure, the bias minimum is more of a kink than a deep
trough, as discussed in Fong & Han (2021). This flattening of the bias profile
at high masses means the depletion radius is easier to measure in lower mass
halos, contrary to the splashback radius.

To avoid this problem, the minimum value of the bias profile itself could be
used as the cosmological test, instead of the radius at which it reaches a
minimum. However, the minimum bias is highly degenerate with the unknown value
of the matter density and the poorly constrained shape of the matter-matter
correlation function. For these reasons, we focus on the depletion radius,
though a more sophisticated analysis that leaves $\rho_{\rm m}$ and $\xi_{\rm
mm}$ free in the fitting could consider the minimum bias value.

We also note that another feature of the depletion zone is the inner depletion
radius, $r_{\rm id}$, which is defined as the radius of maximum mass inflow
rate (Fong & Han, 2021). To determine this radius observationally, peculiar
velocities of galaxies would be required. Since we are considering an analysis
based on weak lensing shear profiles alone, we do not consider this scale
further.

###  3.5 Cosmological dependence

Fig. 2 demonstrates how the splashback and depletion radii vary with mass and
also cosmology. Halos are binned by mass, with a bin width of $0.5\,$dex, and
the mean mass of the bin is plotted on the $x-$axis. The depletion radius is
given by solid green lines, and the splashback radius by dashed blue lines. In
both cases, darker colours correspond to larger values of $S_{8}$. The
simulation results are shown at $z=0$, though in comoving units both radii are
fairly constant with redshift, varying by less than 20% between $z=0$ and
$z=1$ (see Appendix A).

Both radii vary considerably with cosmology, but the cosmological dependence
also changes with halo mass. At low mass, both the splashback and depletion
radii vary smoothly with $S_{8}$, larger $S_{8}$ values corresponding to
smaller radii. For larger mass halos, however, the relationship between these
radii and $S_{8}$ is not as clear. Fig. $2$ of H24 shows that in fact, in the
cluster mass range the splashback radius varies nearly orthogonally to
$S_{8}$.

Figure 2: The scaling of the splashback radius $r_{\rm sp}$ (blue dashed
curves) and the depletion radius $r_{\rm cd}$ (green solid curves) with halo
mass and cosmology, at $z=0$. The radii are determined by fitting the mean 3D
density profile of each mass bin in the simulations. Darker line colours
correspond to larger values of $S_{8}$, with values ranging from $0.572$ to
$1.04$. Also plotted for reference are $2\,r_{200{\rm c}}$ (black dash-dot
line) and $4\,r_{200{\rm c}}$ (black dotted line). Points with error bars
indicate measured values from the literature, the majority of which are of the
splashback radius, with $r_{\rm cd}$ measured in one study (note these are at
a range of redshifts, but have been converted to comoving units where the
expected redshift variation is minimal).

This mass dependence can be traced back to the halo formation time. According
to hierarchical structure formation, galaxy clusters are the most recent bound
structures to form in the universe, and their mean formation time varies
significantly with the background cosmology. For low$-\Omega_{\rm m}$ and
high$-\sigma_{8}$ cosmologies, galaxy clusters typically form at earlier times
(Amoura et al., 2021). Material currently reaching the apocentre of its orbit
at the splashback radius passed through the centre of the cluster potential
some time in the past, a delay we refer to as the "infall time", equal to
$\frac{1}{2}$ of a radial orbital period (Taylor & Babul, 2004). Earlier-
forming clusters will have had a larger mass in place one infall time ago,
compared to later-forming clusters. A larger gravitational potential one
infall time ago produces a larger splashback radius in these systems. For the
depletion radius, earlier-forming clusters have had more time to draw in
material from their environment, pushing the minimum of the bias profile out
to larger radii. These results hint at how profile measurements in the infall
region can provide constraints on cosmological parameters.

In Fig. 2, we also plot the examples to date of directly measured splashback
radii. Masses have been converted to $M_{200{\rm c}}$ using the measured
concentration when it is provided, or the concentration-mass relation of
Ishiyama et al. (2021) when a measured value is not given, and radii have been
converted to comoving units using the mean halo redshift. Current observations
appear consistent with our simulations, except perhaps at the smallest mass
scale.

##  4 Simulated Lensing Profiles

###  4.1 Projected mass density profiles

In weak lensing measurements of mass distributions, a key observable is the
reduced shear of background ‘source’ galaxies, $g$,

$g=\frac{\gamma}{1-\kappa}\,,$ (6)

where $\gamma$ is the shear and $\kappa$ is the convergence. It is often
assumed in the weak lensing limit that $g\approx\gamma$ (Bartelmann &
Schneider, 2001), although $\gamma$ can also be inferred from $g$ through an
iterative reconstruction scheme (Pires et al., 2020). The tangential component
of the shear, $\gamma_{t}(\theta)$, is related to the excess surface mass
density $\Delta\Sigma$ through

$\Delta\Sigma(r)=\gamma_{t}(\theta)\Sigma_{\rm crit}\,,$ (7)

where angular separations $\theta$ on the sky are converted to radial
separations using the angular diameter distance to the lens redshift
$d_{A}(z)$, $r=d_{A}\theta$. The critical surface mass density is given by

$\Sigma_{\rm crit}(z_{\rm l},z_{\rm s})=\frac{c^{2}}{4\pi G}\frac{\chi_{\rm
s}}{\chi_{\rm l}\left(\chi_{\rm s}-\chi_{\rm l}\right)\left(1+z_{\rm
l}\right)}\,.$ (8)

The comoving distance is $\chi(z)$, where the subscript l corresponds to a
foreground lens (a galaxy or a galaxy group$/$cluster), and the subscript s to
a source galaxy whose shape is being measured. $\Sigma_{\rm crit}$ can be
found using knowledge of the source catalogue’s redshift distribution. Note
that both $d_{A}$ and $\Sigma_{\rm crit}$ depend on cosmology. In our
analysis, we will assume a fiducial cosmology in calculating these quantities,
neglecting the cosmological variation. The impact of this approximation on
cosmological constraints will be discussed in more detail, and shown to be
minimal, in Section 6.1.

Figure 3: Left: 3D density profiles are converted into $\Delta\Sigma$
profiles, the excess surface mass density. Middle: The tangential shear
$\gamma_{t}=\Delta\Sigma/\Sigma_{\rm crit}$ is found by assuming a lens
redshift of $0.21$ and UNIONS survey source galaxy properties. Radii are
converted into angles, assuming a fiducial cosmology. Right: Mock
observational data for both simulations using realistic errors and Gaussian
scatter.

Fig. 3 illustrates the steps involved in going from the density profiles
measured in the simulations to a predicted $\gamma$ profile. We show results
for two cosmologies, one with $\Omega_{\rm m}=0.4$, $\sigma_{8}=0.7$ (thin
green line), and another with $\Omega_{\rm m}=0.2$, $\sigma_{8}=1.0$ (thick
magenta line). These correspond roughly to the ends of the ‘banana’, the
typical uncertainty contour derived in previous cluster abundance or cosmic
shear studies.

The stacked density profile is converted into a 2D projected surface mass
density profile using an Abel transform,

$\Sigma(R)=2\int_{R}^{\infty}\frac{\rho(r)r}{\sqrt{r^{2}-R^{2}}}\,dr\,,$ (9)

where $R$ is the projected distance from the centre. Given the surface mass
density distribution, the excess surface mass density around a dark matter
halo $\Delta\Sigma(R)$ is defined as

$\Delta\Sigma(r)=\overline{\Sigma}(R)-\Sigma(R)\,,$ (10)

with

$\overline{\Sigma}(R)=\frac{2}{R^{2}}\int_{0}^{R}R^{\prime}\Sigma(R^{\prime})\,dR^{\prime}\,.$
(11)

This quantity is plotted in the left panel of Fig. 3. Next, using an assumed
lens redshift and the source redshift distribution for a given weak lensing
survey (in this case UNIONS), the $\Delta\Sigma(r)$ profile is converted to a
shear profile, and radial separations are converted to angular separations on
the sky. This gives the $\gamma_{t}(\theta)$ profile in the middle panel.
Finally, realistic uncertainties on the $\gamma$ profile are calculated using
the cluster-lensing-cov package111https://github.com/hywu/cluster-lensing-cov
(Wu et al., 2019) and Gaussian scatter of data points is added to obtain the
mock observations in the right-hand panel. We see a clear difference between
the predicted shear profiles for the two cosmologies, and the amplitude of the
difference greatly exceeds the noise, particularly in the infall region. Thus,
measurements of the infall region with a lensing survey such as UNIONS should
easily distinguish between the two ‘ends of the banana’.

###  4.2 Fitting an observed shear profile

To pick out features in the projected, 2D density profile, we fit the observed
shear profile $\gamma_{t}(\theta)$ using a similar approach to our 3D fitting.
Starting with a model 3D profile given by Eq.( 2, we project and convert this
as described above to generate the corresponding $\gamma_{t}(\theta)$ profile.
We then compare this to the mean profile measured in the simulation using
least squares, and iterate over this process to determine the best choice of
3D parameters. Uncertainties in the fit are propagated along with simulated
measurement uncertainties when generating the results in Section 5. Physical
units are used when stacking the 2D profiles. 20 uniformly spaced data points
are assumed in the fitting range, $r=0.1-10\,r_{\rm pivot}\,$Mpc$\,h^{-1}$.

When fitting the density profiles directly to infer the 3D $r_{\rm sp}$ and
$r_{\rm cd}$ from the simulated halos, $A$ in Eq. (2) is left free, giving the
most freedom to minimise the residual. Allowing $A$ to be free introduces
strong correlation between parameters, however, so when noisy stacked lensing
profiles are being fitted its value is fixed to $\rho_{\rm m}^{\rm Planck}$ to
avoid large parameter uncertainties caused by strong degeneracies, at the cost
of larger residuals.

##  5 Cosmological Constraints from the Infall Region

As demonstrated in H24 and Figs 1–3, the density profile in the infall region
of clusters shows clear systematic variations with cosmology. We will now
investigate the prospects of using this dependence to constrain $\Omega_{\rm
m}$ and $\sigma_{8}$, given lensing data from current and forthcoming surveys.
For each specific survey considered, the uncertainty on shear profiles depends
on the properties of an assumed lens sample (number of halos, mass and
redshift distributions) and source sample (shape noise, surface density, and
redshift distribution). Our assumed source and lens properties are outlined in
Section 5.1. In Sections 5.2 and 5.3 respectively, we then consider
constraints obtained by measuring the depletion radius or the splashback
radius. Finally, in Section 5.4 we estimate constraints based on fitting the
entire infall region.

Throughout this section, covariances are calculated using the cluster-lensing-
cov package (Wu et al., 2019). Uncertainties on the profile are used to
generate Gaussian scatter in the data points. To calculate our final results
(either fitting the profile to extract the splashback/depletion radius, or
comparing the shear profiles directly), we repeat each analysis 50 times with
random realisations of the errors, and average these multiple realisations to
give the final quoted value.

###  5.1 Source and lens catalogue assumptions

We focus on two weak lensing surveys, UNIONS and Euclid Wide. The mass and
redshift bins for the cluster sample assumed in each survey are summarised in
Table 2.

####  5.1.1 UNIONS

UNIONS (Chambers et al., 2020) is a 5-band ground-based imaging survey. The
weak lensing component of the survey will cover an area of $4800\,$deg2, with
source number density of $10\,$deg-2 and shape noise of $0.34$, as described
in Guinot et al. (2022). The source redshift distribution $n(z)$ from that
work is also assumed here. For the lens catalogue, we use the sample described
in Spitzer (2022) as a reference case. This was based on halos in the UNIONS
footprint taken from the catalogue created by Tinker (2021), where a halo
finder was run on SDSS galaxies. However Spitzer (2022) considered only
$2000\,$deg2 of UNIONS, as the weak lensing survey was not complete at that
time. We have assumed that the mean number density of lenses is the same
across the full $4800\,$deg2 of the survey, and therefore lens numbers in that
work are scaled up by a factor of $2.4$. Our three lens mass bins also
correspond to those defined in Spitzer (2022).

####  5.1.2 Euclid

The Euclid Wide survey, planned for the Euclid telescope, is a space-based
survey covering $36\%$ of the sky, providing a total area of $15000\,$deg2,
with a source number density of $30\,$deg-2 and shape noise of $0.3$ (Euclid
Collaboration et al., 2020). The mean redshift of the source distribution is
$\langle z\rangle=0.95$. As lenses, we consider cluster-mass halos detected in
the Euclid photometric survey. The numbers in Bins $1-5$ of Table 2 are
derived from Fig. 3 of Sartoris et al. (2016), assuming a detection threshold
of clusters with $N_{500{\rm c,field}}/\sigma_{\rm field}=5$. The masses are
based on the corresponding selection function in Fig. 2 of that work,
converted to $M_{\odot}\,h^{-1}$ assuming $h=0.7$.

We focus on the lower redshift bins of the survey, to avoid incompleteness
which might bias cluster properties and the inferred shape of the infall
region. We also consider a group-scale sample in Bin 0, assuming a complete
halo sample in the mass range $10^{13}\leq M\,[M_{\odot}\,h^{-1}]\leq
10^{13.5}$ with a total number calculated from the halo mass function of
Despali et al. (2016) for a Planck cosmology. The resulting sample contains
$\sim\\!10^{5}$ halos, and represents the best we could hope to do in this
mass range.

Table 2: Mass and redshift bins assumed for the UNIONS and Euclid-wide weak lensing surveys. Columns give the bin number, mean redshift, mass range and number of clusters respectively. Each redshift shell spans $\pm 0.05$ around the mean value. | UNIONS | Euclid
---|---|---
Bin | $\langle z_{\rm l}\rangle$ | $M/M_{\odot}\,h^{-1}$ | $N_{\rm l}$ | $\langle z_{\rm l}\rangle$ | $M/M_{\odot}\,h^{-1}$ | $N_{\rm l}$
0 | $0.15$ | $10^{13-13.5}$ | $19745$ | $0.2$ | $10^{13-13.5}$ | $10^{5}$
1 | $0.17$ | $10^{13.5-14.0}$ | $9199$ | $0.25$ | $\geq 10^{14.3}$ | $3500$
2 | $0.21$ | $\geq 10^{14.0}$ | $3919$ | $0.35$ | $\geq 10^{14.2}$ | $7000$
3 | — | $0.45$ | $\geq 10^{14.1}$ | $12500$
4 | — | $0.55$ | $\geq 10^{14.0}$ | $20000$
5 | — | $0.65$ | $\geq 10^{14.0}$ | $25000$

###  5.2 Constraints from the depletion radius

The depletion radius is defined as the point where the bias profile reaches a
minimum value. To measure this, we fit the stacked $\Delta\Sigma(R)$ profiles
with the density profile model. Errors on $\Delta\Sigma(R)$ are found using
the cluster-lensing-cov package, as described in the previous section. These
errors are propagated through the fitting procedure. The result is a set of
best-fit parameters for the density profile model, and their covariance. The
covariance is used to generate a set of correlated samples of the model
parameters, assuming Gaussian uncertainties. Then a simple Monte Carlo
operation is performed, drawing from this correlated set of samples
repeatedly, and finding the new depletion radius of the density profile
created on each random draw.

Once a density profile is fitted from a measured $\gamma$ profile, there are
still two unknowns in Eq. (5); the matter density $\rho_{\rm m}$, and the
matter-matter correlation function $\xi_{\rm mm}(r)$, which is calculated
using colossus (Diemer, 2018). We assume a Planck cosmology for both of these,
as in real data we will not know the true value of $\rho_{\rm m}$, and while
$\xi_{\rm mm}(r)$ can be determined from the observations, it requires
knowledge of the galaxy bias. The net effect of this approximation will be
discussed further in Section 6.2.

For the highest mass halos in our sample, there is no actual dip in the bias
profile around the depletion radius, as expected from Fong & Han (2021). These
halos form in high-density environments, where the bias profile remains
positive at all radii. To test the use of the depletion radius as a
cosmological probe, we therefore consider a sample of less massive halos (‘Bin
0’, described above). At present, the depletion radius for halos in this mass
range has been measured through weak lensing by Fong et al. (2022), to a
precision of $\Delta r_{\rm cd}=0.14$.

The top-left and bottom-left panels of Fig. 7 show predicted uncertainties on
the depletion radius in a stacked sample of group-mass halos, with profiles
observed using UNIONS and Euclid respectively. With Euclid data, our predicted
uncertainties lie in the range $0.1$-$0.15$, making them consistent with the
only existing measurement in the literature (Fong et al., 2022). The predicted
uncertainty in this quantity is larger than the cosmological variation over
the range of $\Omega_{\rm m},\sigma_{8}$ considered, as shown in the left-hand
columns of Fig. 7. We conclude that the depletion radius is not a good
candidate for deriving cosmological constraints. The assumptions made when
finding the bias profile also lead to a mismatch between the true and inferred
bias minimum in some cases. This is discussed in more detail in Section 6.2,
and is a serious limitation to using the location of the depletion radius as a
cosmological test.

###  5.3 Constraints from the splashback radius

To determine the splashback radius, we use a similar approach as in Section
5.2. We focus on cluster-mass halos for this measurement, as they will provide
the largest SNR. We note that H24 find that the splashback radius needs to be
measured to an accuracy of $\sim\\!5\%$ if it is to be a competitive
independent probe on $\Omega_{\rm m}$ and $\sigma_{8}$. Other works have shown
it is observable in the future to relatively high precision; for example
(Xhfakaj et al., 2020) find an uncertainty on $r_{\rm sp}$ of $\sim 0.05$
($3-6\%$ depending on the value) from forthcoming weak lensing surveys such as
Euclid and LSST (Ivezić et al., 2019).

The top-middle and bottom-middle panels of Fig. 7 summarise our results for
profiles of stacked cluster-mass samples, measured using lensing data from
UNIONS and Euclid respectively. For each simulation, the value of the
splashback radius and its associated uncertainty is shown. For UNIONS, the
errorbars are too large to constrain deviations from a Planck-like cosmology
(indicated by the magenta band, with the width indicating the measurement
error for that cosmology). The reported values from Planck are $\Omega_{\rm
m}=0.316$ and $\sigma_{8}=0.811$ (Planck Collaboration et al., 2020). On the
other hand for Euclid, depending on the cosmology, we see that significant
deviations from the expected Planck value can be observed. Overall, we find
that deviations from the fiducial Planck values of $\Omega_{\rm m}$ and
$\sigma_{8}$ can be constrained to $\pm 0.05$, using splashback radius
measurements for the Euclid cluster sample.

These results can be further improved by measuring $r_{\rm sp}$ in stacked
clusters from the four other redshift bins in Euclid in Table 2. The
cosmological dependence tends to be similar or larger in higher redshift bins,
and so for each simulation, stacking the SNR from all five bins improves the
constraining power by at least a factor of two, even reaching a factor $4$–$6$
improvement for cosmologies in close proximity to Planck in $\Omega_{\rm
m}$–$\sigma_{8}$. We note this does require complete samples and accurate mass
estimates for clusters at redshifts $\geq 0.3$.

Our results mirror those of Xhfakaj et al. (2020): the projection effects
involved in measuring a $\Delta\Sigma$ profile with weak lensing weaken
constraints on the splashback radius. The quoted uncertainties on $r_{\rm sp}$
in that work are smaller than here, likely due to the different number of
halos assumed ($25\,000$ in their case, compared $3\,500$ in ours). Previous
work has also demonstrated a bias on the inferred splashback radius in
optically selected cluster catalogues (Busch & White, 2017; Sunayama & More,
2019). These selection effects would need to be modelled with more realistic
simulations, depending on the cluster sample assumed.

###  5.4 Constraints from the full profile

Fitting a model of the density profile to observations in order to identify
characteristic radii introduces large uncertainties. To avoid this, we can
consider comparing an observed shear profile directly to simulated profiles,
without assuming any particular analytic form. To evaluate the prospects for
this method, we use the signal-to-noise ratio (SNR) of the difference between
the average shear profiles in the infall region for two different cosmologies.
Given cosmologies $A$ and $B$,

$\displaystyle{\rm SNR}$ $\displaystyle=\frac{\lvert\langle\gamma_{t}({\rm
infall})\rangle_{A}-\langle\gamma_{t}({\rm
infall})\rangle_{B}\rvert}{\sqrt{\sigma_{A}^{2}+\sigma_{B}^{2}}}\,,$ (12)
$\displaystyle\sigma$ $\displaystyle=\frac{1}{N}\sqrt{\sum\sigma_{i}^{2}}\,,$
(13)

where $\sigma_{i}$ is the uncertainty on the $i^{\rm th}$ data point and $N$
is the number of data points in the annulus. The average is computed in an
angular annulus encompassing the infall region, described in the next section.
Before performing this test, we first need to identify the angular scale where
the SNR of the difference is greatest.

####  5.4.1 Scale dependence of the SNR

To test how the SNR of the cosmological variation of the profile depends on
distance from the halo centre, we calculate mean profiles for two cosmologies,
‘$A$’ with $\Omega_{\rm m}=0.25$, $\sigma_{8}=0.85$ and ‘$B$’ with
$\Omega_{\rm m}=0.35$, $\sigma_{8}=0.75$, and take their difference. These two
cases were chosen as they bracket the Planck values in the $\Omega_{\rm
m}$–$\sigma_{8}$ plane along the $S_{8}$ degeneracy direction, the direction
in which the profile of the infall region is expected to vary the most. The
angular dependence of the SNR of this difference, normalised by its maximum
value, is shown in Fig. 4. The SNR depends on the source and lens redshift
distributions, and is shown for two examples, the UNIONS bin with $0.16\leq
z_{\rm l}<0.26$ (thin blue line) and the Euclid bin with $0.2\leq z_{\rm
l}<0.3$ (thick purple line). Solid and dashed-dot lines correspond to group-
mass halos and cluster-mass halos, respectively.

The SNR peaks in the infall region, indicating that this is the best place to
look for differences caused by the cosmological parameters. The smooth peak at
$\sim\\!2$–$4\,\theta_{200{\rm c}}$ persists when comparing differences
between other cosmologies and considering different mass bins, though it does
vary slightly with mass, and typically has a width of $\sim\\!0.15-0.25\,$dex.
To determine the best angular scale to use for a given sample and survey, we
measure the location of the peak in SNR for all combinations of cosmologies
and over a large mass range, and determine the mean $\theta_{\rm peak}$–mass
relation. It is given by

${\rm log}(\theta_{\rm peak}/{\rm deg})=a\,{\rm
log}(M/M_{\odot}\,h^{-1})+b\,,$ (14)

where for UNIONS $a=0.21$, $b=-3.6$, and for Euclid $a=0.36$, $b=-5.9$. The
final annulus over which we integrate has a width of $\pm 0.2\,$dex around the
peak position. For any mass bin, the adopted annulus always encompasses the
peak of SNR, and it is not cosmology dependent. This annulus is used to
produce the results in the top-right and bottom-right panels of Fig. 7, for
UNIONS and Euclid respectively.

Figure 4: The SNR of the difference in stacked shear profiles for two
cosmologies, as a function of angular distance from the halo centre, in units
of the virial radius $\theta_{200{\rm c}}$. For this demonstrative example,
the difference is between the mean shear profiles in cosmology $A$ with
$\Omega_{\rm m}=0.25$, $\sigma_{8}=0.85$ and those from cosmology $B$ with
$\Omega_{\rm m}=0.35$, $\sigma_{8}=0.75$. The example shown uses the UNIONS
bin with $0.16\leq z_{\rm l}<0.26$ (thin blue line) and the Euclid bin with
$0.2\leq z_{\rm l}<0.3$ (thick purple line). Solid and dashed-dot lines
correspond to group-mass halos and cluster-mass halos, respectively. In each
case, the SNR has been normalised by its maximum value.

####  5.4.2 Dependence of the profile variation on mass and redshift

Given the mass-dependent radial annulus described in the previous section, we
calculate the average $\gamma_{t}$ within that annulus for the two different
cosmologies, and determine the SNR of the difference between them. Fig. 5
shows this SNR as a function of mass and redshift, assuming a fixed $1000$
halos per mass and redshift bin, and Euclid weak lensing data.

Figure 5: The SNR of the difference in the predicted mean shear profile
between two cosmologies, $A$ with $\Omega_{\rm m}=0.25$, $\sigma_{8}=0.85$ and
$B$ with $\Omega_{\rm m}=0.35$, $\sigma_{8}=0.75$. The SNR is calculated in
nine redshift bins and fourteen mass bins, with a fixed $1000$ halos per bin.

To get a rough sense of variation of the SNR with mass and redshift, we make a
simple bi-linear fit to the values in Fig. 5:

$\displaystyle f({\rm log}\left(M/M_{\odot}\,h^{-1}\right),z)=$ $\displaystyle
8.93\,{\rm log}\left(M/M_{\odot}\,h^{-1}\right)+74.7\,z$ (15)
$\displaystyle-5.79\,{\rm log}\left(M/M_{\odot}\,h^{-1}\right)\,z-120\,.$

An inverse weighting of $1/$SNR is used in the fit to prioritise the high-SNR
regime. More complicated functions could better fit both high and low SNRs,
but here we opt for this form for simplicity. The fit and residuals are shown
in Fig. 6

Figure 6: Bi-linear fit to the SNR in Fig. 5 (dotted line), together with
residuals with respect to this fit (stars).

Ignoring systematic errors and biases, and assuming the SNR scales as
SNR$\,\propto\sqrt{N}$, we can use these results to set requirements on the
sample size needed to distinguish cosmological models. For example, to
distinguish between the two cosmologies considered so far ($A$ with
$\Omega_{\rm m}=0.25$, $\sigma_{8}=0.85$ and $B$ with $\Omega_{\rm m}=0.35$,
$\sigma_{8}=0.75$) at $5\sigma$ significance, assuming clusters with a mean
mass of $10^{14}M_{\odot}\,h^{-1}$ at $\langle z_{\rm l}\rangle\\!\sim\\!0.2$,
we would need a sample size of $\sim\\!1500$ clusters.

###  5.5 Summary

Fig. 7 summarises the expected accuracy of cosmological constraints derived
using the three methods, depletion radius measurements, splashback radius
measurements, or full profile measurements, for the UNIONS survey (top panels)
and the Euclid Wide survey (bottom panels). The vertical ordering of
cosmologies in each panel is based on the fitting function in Fig. 6 of H24.
To derive these results, we have considered only the lenses in the highest
mass Bin of Table 2 for each survey. Using additional bins would improve these
constraints further.

Figure 7: A summary of our results for two weak lensing surveys, UNIONS (top
three panels), and Euclid (bottom three panels). From left to right, we show
predictions for the measured characteristic depletion radius $r_{\rm cd}$, the
splashback radius $r_{\rm sp}$, and the amplitude of the shear profile in the
infall region for 19 different combinations of $\Omega_{\rm m}$ and
$\sigma_{8}$. In each case, the value measured in the combination of
parameters most similar to a Planck cosmology, $\Omega_{\rm m}=0.3$ and
$\sigma_{8}=0.8$, is highlighted with a purple band. The true 3D $r_{\rm sp}$
and $r_{\rm cd}$ is overlaid with a green symbol for each simulation.

Fig. 8 plots the results of the top-right and bottom-right panels of Fig. 7 in
the $\Omega_{\rm m}$–$\sigma_{8}$ plane for UNIONS and Euclid respectively,
this time finding the SNR of the difference of each cosmology with the
fiducial combination $\Omega_{\rm m}=0.3$, $\sigma_{8}=0.85$. The SNR contours
indicate how well a cosmology in this plane could be constrained from the
fiducial case. Also overlaid are recent results from the SRG$/$eROSITA all-sky
survey Ghirardini et al. (2024), demonstrating the complementary degeneracy
direction of constraints using the infall region.

Figure 8: SNR with which deviations from a fiducial cosmology could be
detected with the full profile measurement method, assuming lensing
measurements from UNIONS (left-hand panel) and Euclid (right-hand panel). In
each case, only the single most massive bin of clusters is used. The fiducial
cosmology is taken to be $\Omega_{\rm m}=0.3$, $\sigma_{8}=0.85$. Approximate
$1$ and $2$–$\sigma$ abundance constraints from Ghirardini et al. (2024) are
shown for comparison (green ellipses).

We can also consider each of the five (three) bins for Euclid (UNIONS) in
Table 2 as independent tests, and sum their resulting SNRs in quadrature. For
UNIONS, the improvement is marginal as the lower mass bins are noisier, and
the cosmological dependence of their infall region is no longer orthogonal to
$S_{8}$. On the other hand, for Euclid, the improvement in the SNR leads to a
reduction in the contour area by a further factor of $\sim\\!2$. The exact
change in area is slightly uncertain, due to the interpolation between limited
numbers of simulations (with parameters indicated by the open circles in Fig.
8). Taken at face value, however, this measurement should reduce the area of
the contour reported by eROSITA by a factor of more than $3$.

##  6 Practical Challenges

###  6.1 Use of a fiducial cosmology in the lensing calculations

Converting the observed shear profile $\gamma_{t}(\theta)$ into a projected
$\Delta\Sigma(r)$ profile involves assuming a fiducial cosmology to convert
angles and redshifts into distances. An incorrect choice of fiducial cosmology
when converting angles to transverse distances causes an overall shift in
scale; an incorrect choice of cosmology when calculating $\Sigma_{\rm crit}$
causes an overall shift in amplitude. Over the range of $\Omega_{\rm m}$
considered, the variation in the angular diameter distance with cosmology
causes a $\pm 1.5\%$ shift in scale relative to Planck. Similarly, the
variation of $\Sigma_{\rm crit}$ causes a $\pm 2.5\%$ change in the amplitude
of the $\Delta\Sigma$ profile recovered, relative to Planck. Fortunately,
these are competing effects. Assuming too high a value of $\Omega_{\rm m}$
when converting from $\gamma_{t}(\theta)$ to $\Delta\Sigma(r)$ would shift the
profile to smaller $r$ and larger amplitude compared to the truth. Profiles
from all simulations would be shifted in the same way, by different amounts
depending on the value of $\Omega_{\rm m}$ relative to the fiducial case.
Considering that the intrinsic variation in the profiles caused by cosmology
is in a direction perpendicular to this shift, these two effects in
combination largely cancel each-other out, leading to a $\leq 1\%$ impact on
the resulting $\Delta\Sigma$ profile caused by a large mis-estimate of the
true $\Omega_{\rm m}$. Thus, we do not include this correction in the main
results.

###  6.2 Measuring the correlation function $\xi_{\rm mm}$

To derive the bias profile corresponding to a given density profile, the
matter-matter correlation function is required in Eq. (5). This function is
itself cosmology-dependent. It can be inferred from observational measurements
of the galaxy correlation function, but then a model for the galaxy bias,
connecting the distribution of galaxies to the distribution of matter, is
needed. The galaxy bias is itself uncertain, and therefore including it
introduces large uncertainties on the recovered cluster bias profile.
Alternatively, instead of assuming a fixed $\xi_{\rm mm}(r)$, a cosmology
dependent $\xi_{\rm mm}$ found using a halo model (or directly from
simulations) could be included in the cosmological inference. This removes the
need to estimate it from observation, but introduces extra freedom in the
fitting procedure. Another consideration is that applying the same $\xi_{\rm
mm}$ to the data and the simulation during the cosmological inference to
generate a bias profile means we are essentially comparing the fitted density
profile to a simulated density profile. This is similar to the method proposed
in Section 5.4, but includes the extra step of fitting the measured shear to
get a density profile.

In the results presented in Section 5.2, we naively assume the correlation
function $\xi_{\rm mm}(r)$ for a Planck-like cosmology when deriving the bias
profile, and as a result, the true location of the bias minimum is not always
recovered. Variations in $\xi_{\rm mm}$ are $\pm 10\%$ in the infall region as
$\sigma_{8}$ is varied by by $\pm 0.05$. The variation is not constant with
radius, so it changes the shape of the bias profile in the infall region,
impacting the recovered depletion radius. Changing $\Omega_{\rm m}$ has a
smaller but non-negligible impact. From this, we conclude that using the
depletion radius (or minimum of the bias) as a cosmological probe is
significantly hampered by uncertainty in the matter-matter correlation
function, and thus probably not competitive with splashback or shear profile
methods.

###  6.3 Cluster mass calibration

Mass calibration of galaxy clusters is a well known challenge. Accurate
determinations of the mean mass and full mass distribution of the stacked halo
sample are crucial in order to define equivalent samples of halos to compare
to in simulations. Previous studies have shown there is a bias when using
optically selected clusters—observed splashback radii are smaller than the
values predicted by simulations (Murata et al., 2020; Giocoli et al., 2024).
Euclid will take advantage of the weak lensing and spectroscopic data
available for its mass calibration, as well as cross-correlating with data
from other surveys (Sartoris et al., 2016). Furthermore, since we only
consider the most massive clusters at low redshift, we can expect fairly
complete samples.

Nonetheless, Fig. 14 of Euclid Collaboration et al. (2024) shows that, in the
redshift range of interest, average weak lensing cluster mass estimates may be
biased low by $5$-$10\%$. We can investigate the impact of this bias by
recomputing mean density and shear profiles for halos with $M\geq 0.9\times
10^{14.3}$ and $M\geq 0.95\times 10^{14.3}$, and comparing to the previous
results obtained with $M\geq 10^{14.3}$, thus measuring how an uncorrected
bias of $5-10\%$ would affect our final constraints. We find that the
difference in the splashback radius in the stacked halo sample is typically
$\ll 1\sigma$ in the case where all halo masses are biased low by $5\%$, where
$\sigma$ is the uncertainty on the inferred value of $r_{\rm sp}$ from weak
lensing profiles. When all halos are biased low by $10\%$, the difference is
larger but still remains below the $1\sigma$ level. When comparing the
amplitude of the stacked shear profile to simulations, errors are much smaller
and the requirements on the mass calibration are more stringent. A downward
mass bias of $10\%$ ($5\%$) for all halos reduces the lensing signal in Euclid
clusters by $7\%$ ($3\%$), a difference that is significantly above the noise.
While a mass bias of this amplitude is probably pessimistic, these results
demonstrate the importance of accurate mass calibration.

Furthermore, the mass uncertainty also has a significant impact. Halos with
masses just below an imposed mass limit can be scattered into the sample, and
those just above scattered out of the sample. The overall effect is a sample
that contains less massive halos than expected. To test the impact, we
generate 5 realisations of Euclid Bin 1 and UNIONS Bin 1 in Table 1 with
masses scattered from their true values assuming a mass uncertainty of
$0.2\,$dex (Old et al., 2015; Li et al., 2021). Results from these
realisations are then compared to the original sample. As expected, the mean
masses of the scattered realisations are each biased low by $\sim\\!0.1\,$dex,
and the measured splashback radii are biased low by $\sim\\!1\sigma$. The
shear profile amplitude is reduced by $10-15\%$ in the infall region. These
results further highlight the importance of accurate mass calibration,
especially when directly comparing the shear profiles to simulations. An
improved method would include realistic mass scatter in the simulated halos
when comparing to observations, such that the main bias was accounted for.
More sophisticated analysis methods, such as including abundance information
by stacking a sample of the $N$ most massive clusters within a survey volume,
or only selecting relaxed clusters (Lau et al., 2013; Gianfagna et al., 2021),
might also help reduce or avoid mass calibration problems. We will consider
this possibility in future work.

###  6.4 Mis-centering

To assess the potential impact of stacking mis-centered halos on our study, we
adopt the framework of George et al. (2012). The authors use a probability
distribution for the size of the offset given by

$P(R_{\rm off})=\frac{R_{\rm off}}{\sigma_{\rm off}}\exp\left(-\frac{R_{\rm
off}^{2}}{2\sigma_{\rm off}^{2}}\right)\,.$ (16)

Using a sample of galaxy groups detected in XMM-Newton (Hasinger et al., 2007)
and Chandra (Elvis et al., 2009) observations, with $\Delta\Sigma$ profiles
determined from COSMOS weak lensing measurements (Leauthaud et al., 2007),
they find a best fit for the width of the offset distribution of $\sigma_{\rm
off}=50\,$kpc. If we assume offsets are randomly drawn from this distribution
for each individual system in a stacked halo sample, compute the modified
$\Delta\Sigma(R)$ profile, and compare it to the case with no offsets, this
will give an idea of the impact of mis-centering.

We assume that every halo has an offset drawn from this distribution, a
pessimistic choice corresponding to the worst-case scenario. For example, in
the optically selected redMaPPer cluster sample, only $\sim\\!25\%$ of
clusters were mis-centered (Zhang et al., 2019). Using $\sigma_{\rm
off}=50\,$kpc, the difference in the amplitude of the shear profile can be up
to $1$–$2\%$ in the infall region, and the impact on the recovered splashback
radius can be a shift of $\sim\\!0.05\,$Mpc$/h$. Both of these results are
within the $1\sigma$ errors for Euclid shear measurements, and are sub-
dominant for UNIONS measured profiles. Assuming $\sigma_{\rm off}$ can be
estimated from observations, as in George et al. (2012), and only a fraction
of clusters are strongly mis-centred, then we do not expect mis-centering to
impact our conclusions. These results corroborate the findings of Zürcher &
More (2019), where they evaluated the impact of mis-centering on the location
of the 3D splashback radius, finding it to increase errors only slightly,
causing shifts of the inferred location well within uncertainties.

##  7 Conclusions

The amplitude and shape of the mean shear profile around galaxy clusters
should depend on cosmology. In universes with low $\Omega_{\rm m}$ and high
$\sigma_{8}$, clusters ‘form’ earlier, assembling their mass into a single
large progenitor at higher redshift relative to universes with high
$\Omega_{\rm m}$ and low $\sigma_{8}$ (Amoura et al., 2021). Material accreted
over the past few Gyr is accelerated by a deeper potential well, and reaches a
larger splashback radius. Thus, the outer density profiles of clusters in low
$\Omega_{\rm m}/$high $\sigma_{8}$ cosmologies will be more extended at the
present day, and the shear signal in these regions will be larger. The
dependence of this effect on $\Omega_{\rm m}$ and $\sigma_{8}$ is close to
orthogonal to the $S_{8}$ degeneracy present in cosmic shear and cluster
abundance studies (H24). Combining lensing measurements of the infall region
around galaxy clusters with cluster number counts can therefore significantly
improve constraints on these parameters, using only low-redshift cluster
properties.

Given realistic assumptions about the cluster samples and lensing data
expected from Euclid, we find that measurements of the splashback radius alone
may constrain deviations from fiducial Planck values of $\Omega_{\rm m}$ and
$\sigma_{8}$ of $\pm 0.05$ or greater. Measuring the ‘depletion radius’, where
the bias reaches a minimum, proves not to be a promising cosmological probe.
This is due to the lack of a pronounced negative dip in the bias at this
radius on cluster scales, and also the need to determine the matter-matter
correlation function in order to calculate the bias. A similar test could in
principle be applied to lower-mass groups or even galaxy halos. In particular,
the depletion radius is a more prominent feature in low-mass dark matter
halos, and thus easier to measure. The density profiles of these lower-mass
systems are more likely to be impacted by baryonic feedback effects, however,
given their lower energy scale, as demonstrated by small-scale lensing or
clustering studies (e.g. Huang et al., 2019). Using the minimum bias value,
instead of the radius at which the bias reaches a minimum, might improve the
prospects of the depletion radius as a cosmological probe for galaxy clusters,
although it requires a good knowledge of the matter density of and the matter-
matter correlation function.

Furthermore, we find that variation of infall region features with cosmology
in these halos is no longer orthogonal to $S_{8}$ as with galaxy clusters, but
has a similar degeneracy direction to $S_{8}$. This limits their utility in
improving abundance or lensing-based constraints. Based on these arguments and
our exploration of the SNR of differences in shear profiles, we can conclude
the best halos to use for a cosmological analysis are high-mass clusters at
low redshift.

Fitting analytic forms to observed shear profiles to infer $r_{\rm sp}$ or
$r_{\rm cd}$ is also a slightly noisy process, reducing the overall SNR of
these tests. By applying a simple angular filter to the mean shear profiles in
the infall region for our simulated clusters, and studying the intrinsic
variation of the mean shear with cosmology, we demonstrated that direct
measurements of the profile in weak lensing surveys such as UNIONS or Euclid
can better constrain $\Omega_{\rm m}$ and $\sigma_{8}$, reaching a final
precision comparable to that of cluster abundance studies, but with a
different degeneracy direction in the $\Omega_{\rm m}$–$\sigma_{8}$ plane.
These constraints could be further improved by using an optimal weighted
filter to extract the largest SNR from the infall region (e.g. Hamana et al.,
2004).

The real power comes from combining constraints based on measurements of the
density profile with traditional results based on cluster abundance. We
estimate that combining the two could reduce the area of the contours reported
by the SRG$/$eROSITA All-Sky Survey by a factor of $1.2$, using one mass and
redshift bin from the UNIONS survey, or a factor of $3$, using five redshift
bins from the Euclid survey. Overall, using the full range of cluster mass and
redshift available, lensing measurements of the infall zone should provide a
competitive, independent cosmological test that is highly complementary to
other low-redshift tests of cosmology, and requires only the data already
collected for abundance and cosmic shear studies. In a follow-up work we will
attempt the first such analysis using clusters in the UNIONS footprint.

## Acknowledgements

We thank the members of the UNIONS collaboration and D. Rana for helpful
feedback in the preparation of this manuscript. C. T. M. is funded by a
Leverhulme Trust Study Abroad Scholarship. J. E. T. acknowledges support from
the Natural Sciences and Engineering Research Council of Canada (NSERC),
through a Discovery Grant. This research was enabled in part by support
provided by Compute Ontario (www.computeontario.ca) and the Digital Research
Alliance of Canada (alliancecan.ca). The python packages numpy, scipy,
matplotlib, lmfit, colossus and cluster-lensing-cov have been used in this
work. For the purpose of open access, the authors have applied a Creative
Commons Attribution (CC BY) licence to any Author Accepted Manuscript version
arising from this submission.

## Data Availability

The simulations used in this article will be shared on reasonable request to
the corresponding author.

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##  Appendix A Variation of the characteristic radii with redshift

Although it is tangential to our main argument, it is also interesting to
study how the cosmology dependence of the characteristic radii varies with
redshift. Fig. 9 shows the redshift dependence of the splashback and depletion
radii for a sub-sample of cosmologies. We see that in comoving units, there is
relatively little variation ($\leq$ 20%) between $z=0$ and $z=1$. This agrees
with other works, (e.g. Diemer et al., 2017; O’Neil et al., 2021), where the
authors find any variation of the splashback radius with redshift is caused by
the variation of $\Omega_{\rm m}(z)$, meaning that little variation is
expected in comoving units.

Figure 9: The mean splashback (left) and depletion (right) radii in comoving
units, for samples in the mass ranges indicated, as a function of redshift and
cosmology.

##  Appendix B Dependence of errors on cosmology

Due to the presence of the matter power spectrum in the covariance
calculation, the predicted errors are in principle cosmology dependent. In our
analysis, we make the approximation that the errors are equal to those of a
fiducial Planck-like cosmology with $\Omega_{\rm m}=0.316$ and
$\sigma_{8}=0.811$ (Planck Collaboration et al., 2020). Fig. 10 shows how the
actual uncertainties depend on cosmology, for Euclid Bin 1 (solid lines) and
UNIONS Bin 2 (dashed lines) defined in Table 2. The uncertainty is plotted as
a function of $\Omega_{\rm m}$ and $\sigma_{8}$, with darker line colours
corresponding to larger values of $S_{8}$. The variation is seen to be
relatively small in the infall region, so we have neglected this correction in
our calculations. In practice there are ways of dealing with this cosmology
dependence in a likelihood analysis (e.g. Eifler et al., 2009), thereby
slightly improving the accuracy of the SNR calculations.

Figure 10: Error on $\Delta\Sigma(R)$ as a function of combinations of
cosmology. Dashed lines correspond to clusters observed in UNIONS, and solid
lines to clusters observed in Euclid. Darker line colours correspond to larger
values of $S_{8}$.


#  A question on the Cauchy problem in the Gevrey classes for weakly
hyperbolic equations

Tatsuo Nishitani††Department of Mathematics, Osaka University:
<EMAIL_ADDRESS>
###### Abstract

For a homogeneous polynomial $p$ in $\xi\in{\mathbb{R}}^{n}$ with Gevrey
coefficients, it is known that the Cauchy problem for any realization of $p$
is well-posed in the Gevrey class of order $s<2$ if the characteristic roots
are real. In this note, we give examples showing the situation of the converse
direction, in particular the optimality of the Gevrey order $s=2$.

##  1 Introduction

Consider a polynomial in $\xi=(\xi_{1},\ldots,\xi_{n})$ of degree $m$ with
Gevrey coeffcietnts

$P(x,\xi)=\sum_{|\alpha|\leq
m}a_{\alpha}(x)\xi^{m}=p(x,\xi)+\sum_{j=0}^{m-1}P_{j}(x,\xi)$

where $a_{\alpha}(x)$ are in some Gevrey classes $\gamma^{(s)}(\Omega)$ or
$\gamma^{\langle{s}\rangle}(\Omega)$ defined in a neighborhood of the origin
of ${\mathbb{R}}^{n}$ and $p(x,\xi)$, $P_{j}(x,\xi)$ denotes the homogeneous
part of degree $m$ and $j<m$ in $\xi$ respectively. Gevrey class
$\gamma^{(s)}(\Omega)$ is the set of functions $f\in C^{\infty}(\Omega)$ such
that for any compact set $K\Subset\Omega$ there exist constants $C>0,A>0$ for
which the following inequalities hold:

(1.1) $|D^{\alpha}f(x)|\leq CA^{|\alpha|}(|\alpha|!)^{s},\;\;x\in
K,\;\;\alpha\in{\mathbb{N}}^{n}.$

For a given $P(x,\xi)$ we consider its several realizations (quantizations) as
differential operators. We define ${\rm op}^{t}({P})$, $0\leq t\leq 1$ by

$({\rm op}^{t}({P})u)(x)=(2\pi)^{-n}\int
e^{i(x-y)\xi}P((1-t)x+ty,\xi)u(y)dyd\xi.$

Note that, assuming that $a_{\alpha}(x)$ are constant outside some compact
neighborhood of the origin for simplicity, we see

$\sum_{|\alpha|\leq m}a_{\alpha}(x)D^{\alpha}u(x)={\rm
op}^{0}({P})u(x),\quad\sum_{|\alpha|\leq
m}D^{\alpha}\big{(}a_{\alpha}(x)u(x)\big{)}={\rm op}^{1}({P})u(x).$

When $t=1/2$ the quantization ${\rm op}^{1/2}({P})$ is called Weyl
quantization and also denoted by ${\rm op}^{w}({P})$. Note that

$\lim_{\lambda\to\infty}\lambda^{-m}e^{-\lambda x\xi}{\rm
op}^{t}({P})e^{\lambda x\xi}=p(x,\xi)$

so that the principal symbol of ${\rm op}^{t}({P})$ is independent of
realization and given by the highest homogeneous part of $P(x,\xi)$.

Denote $x=(x_{1},x_{2},\ldots,x_{n})=(x_{1},x^{\prime})$ and consider the
Cauchy problem

(1.2) $\left\\{\begin{array}[]{ll}{\rm
op}^{t}({P})u(x)=0,\quad(x_{1},x^{\prime})\in\omega\cap\\{x_{1}>\tau\\},\\\\[8.0pt]
D_{1}^{j}u(0,x^{\prime})=u_{j}(x^{\prime}),\;\;j=0,\ldots,m-1,\;\;x^{\prime}\in\omega\cap\\{x_{1}=\tau\\}\end{array}\right.$

where $\omega\subset\Omega$ is some open neighborhood of the origin of
${\mathbb{R}}^{n}$. The Cauchy problem (1.2) is (uniformly) well-posed in
$\gamma^{(s)}$ near the origin if there exist $\omega$ and $\epsilon>0$ such
that for any $u_{j}\in\gamma^{(s)}({\mathbb{R}}^{n-1})$ and for any
$|\tau|<\epsilon$ the Cauchy problem (1.2) has a unique solution $u\in
C^{m}(\omega)$. We say that the Cauchy problem is locally solvable in
$\gamma^{(s)}$ at the origin if for any
$u_{j}\in\gamma^{(s)}({\mathbb{R}}^{n-1})$ one can find a neighborhood
$\omega$ of the origin, which may depend on $\\{u_{j}\\}$, such that (1.2)
with $\tau=0$ has a solution $u\in C^{m}(\omega)$. We assume that the
hyperplanes $x_{1}=\text{const.,}$ are non-characteristic, that is

(1.3) $p(x,\theta)\neq 0,\;\;\theta=(1,0,\ldots,0),\;\;x\in\Omega,$

which is almost necessary for $C^{\infty}$ well-posedness of the Cauchy
problem ([8]). Then without restrictions one may assume
$a_{(m,0,\ldots,0)}(x)=1$. If the Cauchy problem (1.2) is locally solvable in
$\gamma^{(s)}$, $s>1$ at the origin then $p(0,\xi_{1},\xi^{\prime})=0$ has
only real roots for any $\xi^{\prime}\in{\mathbb{R}}^{n-1}$ ([9]) so we assume
without restrictions

(1.4) $p(x,\xi-i\theta)=0,\;\;\xi\in{\mathbb{R}}^{n-1},\;\;x\in\Omega.$

The next results are implicit in [3] and [5].

######  Theorem 1.1.

Assume $P(x,\xi)=p(x,\xi)+\sum_{j=0}^{r}P_{j}(x,\xi)$ and the coefficients
$a_{\alpha}(x)$ belong to $\gamma^{(m/r)}$. Then for any $0\leq t\leq 1$ the
Cauchy problem for ${\rm op}^{t}({P})$ is well-posed in $\gamma^{(s)}$ near
the origin for $1<s<\min{\\{m/r,2\\}}$.

######  Corollary 1.1.

Assume that the coefficients $a_{\alpha}(x)$ belong to $\gamma^{(2)}$. Then
for any $0\leq t\leq 1$ the Cauchy problem for ${\rm op}^{t}({p})$ is well-
posed in $\gamma^{(s)}$ near the origin for $1<s<2$.

To confirm the results, note that under the assumption there is some $M>0$
such that

(1.5) $P(x,\xi+i\tau\theta)\neq 0,\quad|\tau|\geq M(1+|\xi|)^{r/m},\quad
x,\;\xi\in{\mathbb{R}}^{n}$

that is $P$ is $m/r$ \- hyperbolic (see [6]). Then Theorem 1.1 was proved for
${\rm op}^{0}({P})$ in [5] and Corollary 1.1 for ${\rm op}^{0}({p})$ is
implicit in [3]. Next, we recall a formula for change of quantization (e.g.
[7]). One can pass from any $t$\- quantization to the $t^{\prime}$\-
quantization by

(1.6) ${\rm op}^{t^{\prime}}({a_{t^{\prime}}})={\rm op}^{t}({a_{t}}),\quad
a_{t^{\prime}}(x,\xi)=e^{-i(t^{\prime}-t)D_{x}D_{\xi}}a_{t}(x,\xi)$

for $a(x,\xi)\in S^{m}_{1,0}$, symbols of classical pseudodifferential
operators. In particular, we have

${\rm op}^{t}({P(x,\xi)})={\rm
op}^{0}\big{(}e^{itD_{x}D_{\xi}}P(x,\xi)\big{)}.$

On the other hand, from [3, Proposition 3] one has

$\big{|}(\partial_{x}^{\beta}\partial_{\xi}^{\alpha}p)(x,\xi-i\theta)\big{|}\leq
C_{\alpha\beta}(1+|\xi|)^{|\beta|}|p(x,\xi-i\theta)|,\quad\alpha,\beta\in{\mathbb{N}}^{n}$

which is sufficient to estimate new terms that appear by operating
$e^{itD_{x}D_{\xi}}$ to $P(x,\xi)$.

##  2 A question on Theorem 1.1

If $P$ is of constant coefficients, $P(x,\xi)=P(\xi)$, the Cauchy problem for
$P(D)$ is $\gamma^{(s)}$ well-posed for $1<s<m/r$ ([6]) if (1.5) holds where
it is understood that the Cauchy problem is $C^{\infty}$ well-posed when
$r=0$, which corresponds to the hyperbolicity in the sense of Gårding ([4]).
It is clear that the results are optimal considering examples
$P(D)=D_{1}^{m}+cD_{n}^{r}$ with a suitable $c\in{\mathbb{C}}$. In the
variable coefficient case, on the other hand, we are restricted to
$1<s<\min{\\{m/r,2\\}}$ in both Theorem 1.1 and Corollary 1.1. Here we give an
example showing that one can not exceed $2$, at least when $m\geq 3$. Consider

(2.1)
$P_{b}(x,\xi)=\xi_{1}^{3}-(\xi_{2}^{2}+x_{2}^{2}\xi_{n}^{2})\xi_{1}-b\,x_{2}^{3}\xi_{n}^{3}$

with $b\in{\mathbb{R}}$ which was studied in [2]. Note that (1.4) is
equivalent to $b^{2}\leq 4/27$. In [2] it was proved that there is
$0<{\bar{b}}<2/3\sqrt{3}$ such that the Cauchy problem for ${\rm
op}^{0}({P_{\,{\bar{b}}}})$ is not locally solvable at the origin in
$\gamma^{(s)}$ for $s>2$.

######  Proposition 2.1.

Let $m\geq 3$ and $n\geq 3$ and consider

$p(x,\xi)=\xi_{1}^{m-3}\big{(}\xi_{1}^{3}-(\xi_{2}^{2}+x_{2}^{2}\xi_{n}^{2})\xi_{1}-{\bar{b}}\,x_{2}^{3}\xi_{n}^{3}\big{)}$

which is a homogeneous polynomial in $\xi$ of degree $m$ with polynomial
coefficients. For any $0\leq t\leq 1$ the Cauchy problem for ${\rm
op}^{t}({p})$ is not locally solvable at the origin in $\gamma^{(s)}$ for
$s>2$, in particular, ill-posed in $\gamma^{(s)}$, $s>2$ near the origin.

From (1.6) one sees that ${\rm op}^{t^{\prime}}({p})={\rm op}^{t}({p})$ for
any $0\leq t^{\prime},t\leq 1$ so that

${\rm
op}^{t}({p})=D_{1}^{m-3}\big{(}D_{1}^{3}-(D_{2}^{2}+x_{2}^{2}D_{n}^{2})D_{1}-{\bar{b}}\,x_{2}^{3}D_{n}^{3}\big{)}=D_{1}^{m-3}{\rm
op}^{0}({P_{\,\bar{b}}}).$

For any given $u_{j}(x^{\prime})\in\gamma^{(s)}$, $j=0,1,2$ we define
$u_{j+1}(x^{\prime})\in\gamma^{(s)}$ by setting
$u_{j+3}=(D_{2}^{2}+x_{2}^{2}D_{n}^{2})u_{j+1}+{\bar{b}}x_{2}^{3}D_{n}^{2}u_{j}$,
$j=0,\ldots,m-4$ successively. Assume that $u\in C^{m}(\omega)$ satisfies
$D_{1}^{m-3}{\rm op}^{0}({P_{\,\bar{b}}})u=0$ with
$D_{1}^{j}u(0,x^{\prime})=u_{j}(x^{\prime})$, $0\leq j\leq m-1$ then $w={\rm
op}^{0}({P_{\,\bar{b}}})$ satisfies $D_{1}^{m-3}w=0$ with
$D_{1}^{j}w(0,x^{\prime})=0$, $j=0,\ldots,m-4$ hence ${\rm
op}^{0}({P_{\,\bar{b}}})u=0$ contradicting with non local solvability of the
Cauchy problem for ${\rm op}^{0}({P_{\,\bar{b}}})$.

Here is a general question why $s=2$ $($independent of $m\geq 3$$)$ ?

A similar phenomenon is observed in the Cauchy problem for uniformly
diagonalizable first order systems ([12, Theorem 3.3]). When $m=2$ we have a
result similar to Proposition 2.1:

######  Proposition 2.2.

Let $n\geq 3$ and consider

$P_{mod}(x,\xi)=\xi_{1}^{2}-2x_{2}\xi_{1}\xi_{n}-\xi_{2}^{2}-x_{2}^{3}\xi_{n}^{2}$

which is a homogeneous polynomial in $\xi$ of degree $2$ with polynomial
coefficients. For any $0\leq t\leq 1$ the Cauchy problem for ${\rm
op}^{t}({P_{mod}})$ is not locally solvable at the origin in $\gamma^{(s)}$
for $s>5$, in particular, ill-posed in $\gamma^{(s)}$, $s>5$ near the origin.

This result for ${\rm op}^{0}({P_{mod}})$ was proved in [1] (where there is
some insufficient part of the proof, see the correction given in [11]). Then
to conclude Proposition 2.2 it is enough to note ${\rm op}^{t}({P_{mod}})={\rm
op}^{0}({P_{mod}})$ for $0\leq t\leq 1$.

Now we would ask ourselves is there an example of a homogeneous polynomial $p$
in $\xi=(\xi_{1},\xi_{2},\ldots,\xi_{n})$, $n\geq 3$ of degree $2$ with real
analytic coefficients satisfying (1.3) and (1.4) such that the Cauchy problem
for ${\rm op}^{t}({p})$, for any $0\leq t\leq 1$, is ill-posed in
$\gamma^{(s)}$, $s>2$.

For the special case $n=2$ ($m=2$) we have;

######  Proposition 2.3.

Consider

$P(x,\xi)=\xi_{1}^{2}-2a(x)\xi_{1}\xi_{2}+b(x)\xi_{2}^{2}+c(x),\quad
x=(x_{1},x_{2})\in{\mathbb{R}}^{2}$

which is a polynomial in $\xi=(\xi_{1},\xi_{2})$ of degree $2$ without
homogeneous part of degree $1$ with real analytic coefficients
$a(x),b(x),c(x)$ such that $\Delta(x)=a^{2}(x)-b(x)\geq 0$ near the origin
$($$a(x)$, $b(x)$ are assumed to be real$)$. Then the Cauchy problem for ${\rm
op}^{w}({P})$ is $C^{\infty}$ well-posed near the origin.

In fact if we make a real analytic change of coordinates
$y=\kappa(x)=(x_{1},\phi(x))$ such that
$\phi_{x_{1}}(x)-a(x)\phi_{x_{2}}(x)=0$, $\phi(0,x_{2})=x_{2}$ where
$\phi_{x_{j}}=\partial\phi(x)/\partial x_{j}$ we see that

(2.2) $\begin{split}&{\rm op}^{w}({P(x,\xi)})(u\circ\kappa)\\\ =&\Big{(}{\rm
op}^{0}\big{(}\eta_{1}^{2}-\alpha{\tilde{\Delta}}\eta_{2}^{2}+\beta_{1}{\tilde{\Delta}_{x_{2}}}\eta_{2}+\beta_{2}{\tilde{\Delta}}\eta_{2}+\beta_{3}\eta_{1}+\beta_{4}\big{)}u\Big{)}\circ\kappa\end{split}$

where ${\tilde{\Delta}}=\Delta\circ\kappa^{-1}$,
${\tilde{\Delta}_{x_{2}}}={\Delta_{x_{2}}}\circ\kappa^{-1}$ and
$\alpha=\alpha(y)\geq c>0$, $\beta_{i}=\beta_{i}(y)$ are real analytic near
$y=0$. To prove the result, noting that
$\big{|}{\tilde{\Delta}_{x_{2}}}\big{|}\leq
C\big{|}{\tilde{\Delta}_{y_{2}}}\big{|}\leq
C^{\prime}\big{|}\sqrt{{\tilde{\Delta}}}\big{|}$, it suffices to apply [10,
Theorem 1.1] to the right-hand side of (2.2).

We could possibly ask if there exists some ${\bar{s}}>2$ such that for any
homogeneous polynomial $p$ in $\xi$ of degree $2$ with real analytic
coefficients satisfying (1.3) and (1.4) the Cauchy problem for ${\rm
op}^{w}({p})$ is well-posed in $\gamma^{(s)}$ for $s<{\bar{s}}$ near the
origin.

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