# A non-linear solution to the $S_{8}$ tension II: Analysis of DES Year 3
cosmic shear

Calvin Preston1,2, Alexandra Amon 1,2, George Efstathiou 1,2
${1}$ Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3
OHA.
${2}$ Institute of Astronomy. Madingley Road, Cambridge, CB3 OHA. United
Kingdom E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Weak galaxy lensing surveys have consistently reported low values of the
$S_{8}$ parameter compared to the Planck $\Lambda{\rm CDM}$ cosmology. Amon &
Efstathiou (2022) used KiDS-1000 cosmic shear measurements to propose that
this tension can be reconciled if the matter fluctuation spectrum is
suppressed more strongly on non-linear scales than assumed in state-of-the-art
hydrodynamical simulations. In this paper, we investigate cosmic shear data
from the Dark Energy Survey (DES) Year 3. The non-linear suppression of the
matter power spectrum required to resolve the $S_{8}$ tension between DES and
the Planck $\Lambda{\rm CDM}$ model is not as strong as inferred using KiDS
data, but is still more extreme than predictions from recent numerical
simulations. An alternative possibility is that non-standard dark matter
contributes to the required suppression. We investigate the redshift and scale
dependence of the suppression of the matter power spectrum. If our proposed
explanation of the $S_{8}$ tension is correct, the required suppression must
extend into the mildly non-linear regime to wavenumbers $k\sim 0.2h{\rm
Mpc}^{-1}$. In addition, all measures of $S_{8}$ using linear scales should
agree with the Planck $\Lambda{\rm CDM}$ cosmology, an expectation that will
be testable to high precision in the near future.

###### keywords:

cosmology: cosmological parameters, weak lensing, observations,

##  1 Introduction

The six-parameter $\Lambda{\rm CDM}$ cosmological model has been incredibly
successful in explaining the anisotropies of the cosmic microwave background
(CMB; e.g. Planck Collaboration et al., 2020a), baryon acoustic oscillations
(BAO; e.g. Alam et al., 2021) and a wide range of other astronomical data.
Nevertheless, there are indications of ’tensions’ between this model and some
observations. The ‘Hubble tension’, i.e. the discrepancy between distance
ladder measurements of the Hubble parameter at present day, $H_{0}$, and the
value inferred from the CMB assuming the $\Lambda{\rm CDM}$ model, is the most
well known (for reviews see Freedman, 2021; Shah et al., 2021; Kamionkowski &
Riess, 2022). In addition, there is an apparent discrepancy between the
amplitude of the matter fluctuations inferred from the CMB and that measured
in cosmic shear surveys which has become known as the ‘$S_{8}$
tension’111Where $S_{8}=\sigma_{8}(\Omega_{\rm m}/0.3)^{0.5}$, $\Omega_{\rm
m}$ is the present day matter density in units of the critical density and
$\sigma_{8}$ is the root mean square linear amplitude of the matter
fluctuation spectrum., (e.g. Heymans et al., 2013; Asgari et al., 2021; Amon
et al., 2022; Secco et al., 2022; Dalal et al., 2023; Li et al., 2023). Since
new physics may be required to explain these tensions, they have become the
focus of many recent observational and theoretical studies.

This paper is a sequel to Amon & Efstathiou (2022), hereafter Paper I, and is
devoted exclusively to the $S_{8}$ tension. Paper I investigated the
hypothesis that the Planck $\Lambda{\rm CDM}$ cosmology accurately describes
matter fluctuations on linear scales, including their growth rate. The $S_{8}$
tension is then explained by modifying the matter power spectrum on non-linear
scales. Paper I adopted a simple phenomenological model for the matter power
spectrum

$P_{\rm m}(k,z)=P^{\rm L}_{\rm m}(k,z)+A_{\rm mod}[P^{\rm NL}_{\rm
m}(k,z)-P^{\rm L}_{\rm m}(k,z)]\,,$ (1)

where the superscript ${\rm L}$ denotes the linear theory power spectrum and
the superscript NL denotes the non-linear power spectrum in a model in which
the matter behaves like cold dark matter (i.e. ignoring the thermal pressure
of baryons and baryonic feedback). The $A_{\rm mod}$ parameter modulates the
amplitude of the non-linear spectrum and can describe a suppression of power
on small scales. Paper I compared this model to the weak lensing measurements
from the Kilo-Degree Survey (KiDS) reported by Asgari et al. (2021) and showed
that the Planck  $\Lambda{\rm CDM}$ cosmology provides acceptable fits to the
KiDS shear-shear two-point statistics if the suppression parameter has a value
$A_{\rm mod}\approx 0.69$. Our hypothesis explains why the Planck
$\Lambda{\rm CDM}$ cosmology agrees so well with: (a) the background expansion
history measured from Type Ia supernovae over the entire redshift range
$0.1\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}z\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}1.5$ spanned by the Pantheon
sample, (e.g. Brout et al., 2022) and (b) gravitational lensing of the CMB
(Planck Collaboration et al., 2020b; Darwish et al., 2021; Omori et al.,
2023), since these measurements are dominated by linear scales and are
consistent with the predictions of General Relativity (i.e. there is no
evidence for ‘gravitational slip’ as might be expected in theories of modified
gravity, see e.g. Bertschinger 2011). In fact, point (b) has been strongly
reinforced with the recently released CMB weak lensing results from the
Atacama Cosmology Telescope (ACT), which are in excellent agreement with the
Planck  $\Lambda{\rm CDM}$ cosmology (Madhavacheril et al., 2023, and
references therein).

Our hypothesis predicts that all measures of the fluctuation amplitude
dominated by linear scales, such as galaxy redshift-space distortions (e.g.
Alam et al., 2021; Philcox & Ivanov, 2022; Chen et al., 2022a; D’Amico et al.,
2022) and cross-correlations of CMB lensing with galaxy surveys (e.g. Chang et
al., 2023; White et al., 2022; Chen et al., 2022b) should agree with the
predictions of the Planck $\Lambda{\rm CDM}$ model. As discussed in Paper I
there is not yet a consensus on the interpretation of these types of
measurements, with some authors reporting tension with $\Lambda{\rm CDM}$.
This situation is likely to change in the near future, for example, via cross-
correlations of the new ACT lensing maps and galaxy surveys and redshift-space
distortion measurements with the Dark Energy Spectroscopic Instrument, (DESI
Collaboration et al., 2016, DESI). Should such measurements conflict with
$\Lambda{\rm CDM}$ on linear scales, our proposed solution of the $S_{8}$
tension will become untenable.

If the hypothesis of Paper I is correct, what physical processes might be
responsible for a suppression of power on small scales? Baryonic feedback is
an obvious candidate, though the suppression required to explain the KiDS data
is stronger than seen in recent cosmological hydrodynamical simulations (e.g.
Dubois et al., 2014; McCarthy et al., 2017; Springel et al., 2018). The
physics of feedback is, however, complex and is not yet sufficiently well
understood to exclude baryonic feedback as the sole cause of the suppression.
Suppression of the power spectrum can also be achieved by invoking more
complex models of dark matter, for example, adding a component of warm or
axionic dark matter (see e.g. Widrow & Kaiser, 1993; Hu et al., 2000; Hui et
al., 2017; Rogers et al., 2023) to the cold dark matter of $\Lambda{\rm CDM}$.
Since it may be difficult to disentangle the effects of baryonic feedback from
those of exotic dark matter, we adopt a phenomenological approach to modelling
the power spectrum on non-linear scales. An alternative approach, based on ad
hoc modifications to the halo mass function, has been described recently by Gu
et al. (2023). As in Paper I, we remain agnostic as to the exact physical
cause of power spectrum suppression.

In this paper, we analyse the cosmic shear two-point statistics from the Dark
Energy Survey (DES) Year 3 analysis presented in Amon et al. (2023); Secco,
Samuroff et al. (2022), hereafter DES22. The DES measurements are mostly
independent of those from KIDS and have somewhat higher statistical power. Our
analysis of DES, therefore, provides a check of the results of Paper I. In
this work, we extend the analysis of Paper I by generalising Eq. 1 to allow
the power suppression to vary with redshift and wavenumber. As we will see it
is difficult to extract detailed information on these dependencies with the
current generation of weak lensing surveys.

This paper is structured as follows. Sec. 2 discusses the Dark Energy Survey,
the data used and the modelling choices made in our analysis. We compare the
DES and KiDS constraints on the parameter $A_{\rm mod}$ in Sec. 3. Sec. 4
explores constraints imposed by the DES measurements on the redshift
dependence of the power suppression. In Sec. 5, we explore the scale
dependence of the suppression to provide more stringent tests of whether
baryon feedback can resolve the $S_{8}$ tension. Finally, we summarize our
conclusions and discuss the implications of our results in Sec. 6.

##  2 Dark Energy Survey

Figure 1: The DES Y3 $\xi_{\pm}$ cosmic shear measurements of Amon et al.
(2023); Secco, Samuroff et al. (2022) and best-fit theoretical models using
three modelling choices. In red we show the best fit of variant 3: when
‘$\Lambda{\rm CDM}$-Optimised’ scale cuts are applied to mitigate the effects
of baryonic feedback, following DES22 The blue line shows the best fit if all
scales are used and baryonic feedback is ignored. The yellow line denotes the
best fit when all scales are used, the $A_{\rm mod}$ parameter is included and
the cosmology is based on the Planck prior as described in Sec. 2. All three
provide good fits, with very similar values for the reduced-$\chi^{2}$, quoted
in Table 2. Residuals are shown below each panel, offset from one another to
be visible.

The Dark Energy Survey (DES Abbott et al., 2018, 2021) is a six-year imaging
programme that observed $\sim 5000\ $deg2 of the Southern Hemisphere in five
photometric bands. In this paper, we use data from the first three years of
observations (DES Y3) which span the complete footprint of the full survey at
a reduced depth. The DES Y3 cosmic shear two-point correlation function
measurements presented in DES22 are based upon the shape estimations of over
100 million galaxies (Gatti et al., 2022), calibrated using a suite of image
simulations (MacCrann et al., 2022). The data is divided into four redshift
bins and the redshift distributions are calibrated to give an overall mean
redshift of $z=0.63$ (Myles et al., 2021).

We reanalyse the DES cosmic shear data using the public DES pipeline and
following the analysis of DES22222Following the DES Y3 analysis, we use nested
sampling via the Cosmosis cosmology parameter estimation framework (Zuntz et
al., 2015), but instead use the MultiNest sampler (Feroz et al., 2009), as
used in Paper I. For this specific analysis, investigations of a subset of
models made with the sampler Polychord (Handley et al., 2015) gave consistent
results, though requiring substantially more computing time for convergence..
We update the analysis choices as follows: we assume three neutrino species
with two massless states and one massive state with a mass of $0.06\ {\rm
eV}$; we adopt the non-linear alignment (NLA) model for intrinsic alignments
as described in Secco, Samuroff et al. (2022); we use HMCode2020 (Mead et al.,
2021) to model the nonlinear matter power spectrum, which generates a non-
negligible shift in the $S_{8}$ posterior compared to the Halofit model
(Takahashi et al., 2012) used in DES22. These analysis choices are tested and
compared in detail in DES & KiDS Collaborations (2023).

Table 1 lists the priors used in this paper. For the ‘Free’ cases, we applied
uninformative priors on the cosmological parameters, following DES22, with the
exception of the neutrino mass which is kept fixed in our analysis. The
remaining entries in Table 1 give the priors for the redshift calibration for
each bin, $\Delta z$, the shear calibration for each bin, $m$, and the single
intrinsic alignment parameter $A_{1}$ of the NLA model, as in DES22.

In Paper I, which was exploratory in nature, we demonstrated that the best fit
Planck $\Lambda{\rm CDM}$ cosmology provides acceptable fits to the KiDS
shear-shear statistics if $A_{\rm mod}\sim 0.7$. However, we did not account
for any uncertainty on the cosmological parameters, particularly $S_{8}$, as
measured by Planck. As a consequence ,Paper I overestimated the suppression of
small scale power required to resolve the $S_{8}$ tension with KiDS. In this
paper, instead of a joint analysis, we include the Planck uncertainty on
$S_{8}$ by applying a prior on $S_{8}$ and $\Omega_{\rm m}$ derived from the
base $\Lambda{\rm CDM}$ Planck TTTEEE chains:

$\displaystyle\chi^{2}_{\rm Planck}$ $\displaystyle=$ $\displaystyle
19249.9592(S_{8}-0.8275)^{2}$ (2) $\displaystyle-$ $\displaystyle
65748.1012(S_{8}-0.8275)(\Omega_{\rm m}-0.3134)$ $\displaystyle+$
$\displaystyle 71122.5486(\Omega_{\rm m}-0.3134)^{2}.$

This prior is described in more detail in Appendix A. The scalar spectral
index and physical baryon density are poorly determined by DES and KiDS but
are determined to high-precision by Planck  within the context of the six-
parameter $\Lambda{\rm CDM}$ cosmology. Therefore, we fix these parameters to
the best-fit Planck  values from Efstathiou & Gratton (2021): $n_{\rm
s}=0.9671$ and $\omega_{\rm b}=\Omega_{\rm b}h^{2}=0.02226$. Finally, for a
given value of $\Omega_{\rm m}$, the Hubble parameter $h$ is set from the
well-determined parameter combination $\Omega_{\rm m}h^{3}=0.09612$. Applying
a prior in this way allows us to accurately account for the Planck
uncertainty without needing to evaluate the Planck  likelihood in our
analysis.

Table 1: Summary of parameters and their priors used in the analysis. The prior ranges for parameters with flat priors and given in square brackets with the prefix F. The mean and $1\sigma$ width for parameters with Gaussian priors are given by the brackets with prefix $G$. The prior ranges listed here are those used in the ‘free’ cosmology analysis throughout this paper. The third column lists the fiducial Planck TTTEEE $\Lambda{\rm CDM}$ parameters for the 10.5HM likelihood as given in Efstathiou & Gratton (2021). In this paper, when we apply a Planck prior, we fix the cosmological parameters that are unconstrained by weak lensing to the fiducial Planck values, but we vary $\Omega_{\rm m}$ and $S_{8}$ using the prior defined in Eq. 2. For intrinsic alignments, we chose to use the non-linear alignment model (NLA) for this analysis using an uninformative prior for the tidal alignment parameter $a_{1}$. We use the data calibration parameters recommended in the fiducial DES analysis Abbott et al. (2021). When using the Planck prior, the values of the nuisance parameters are fixed to their best-fit value from the DESY3 analysis. Parameter | ‘Free’ prior | ‘Planck ’
---|---|---
Cosmological |  |
$\Omega_{\rm m}$ Total matter density | F$[0.1,0.9]$ | -
$\Omega_{\rm b}$ Baryon density | F$[0.03,0.07]$ | -
$\Omega_{\rm b}h^{2}$ | - | 0.02226
$10^{-9}A_{\rm s}$ Scalar spectrum amp. | F$[0.5,5.0]$ | -
$h$ Hubble parameter | F$[0.55,0.91]$ | -
$\Omega_{\rm m}h^{3}$ | - | 0.09612
$n_{\rm s}$ Spectral index | F$[0.87,1.07]$ | 0.9671
$m_{\nu}$ Neutrino mass $({\rm eV})$ | $0.06$ | 0.06
Systematic |  |
$A_{1}$ Intrinsic alignment amp. | F$[-5,5]$ | 0.131
$\Delta z^{1}$ Source redshift 1 | $G(0.0,0.018)$ | 0.004
$\Delta z^{2}$ Source redshift 2 | $G(0.0,0.015)$ | -0.001
$\Delta z^{3}$ Source redshift 3 | $G(0.0,0.011)$ | 0.001
$\Delta z^{4}$ Source redshift 4 | $G(0.0,0.017)$ | 0.001
$m^{1}$ Shear calibration 1 | $G(-0.006,0.009)$ | -0.006
$m^{2}$ Shear calibration 2 | $G(-0.020,0.008)$ | -0.019
$m^{3}$ Shear calibration 3 | $G(-0.024,0.008)$ | -0.025
$m^{4}$ Shear calibration 4 | $G(-0.037,0.008)$ | -0.037

Results of analysing the DES Y3 $\xi_{\pm}$ data for the different
parameterisations and modelling choices discussed in this paper are summarized
in Table 2. The first two entries (labelled 1 and 2), summarize results from
DES22 (see their Table III) using their ‘$\Lambda{\rm CDM}$ Optimised’ scale
cuts, which excludes data points lying within the shaded regions in Fig. 1
from the analysis. Both of these analysis variants use Halofit to model the
non-linear power spectrum. Variant 1 uses the 5-parameter Tidal Alignment and
Tidal Torquing (TATT) model for intrinsic alignments and allows the neutrino
mass to vary, as described in DES22. In variant 2, the neutrino mass is fixed
to $0.06\ {\rm eV}$ and the two-parameter NLA model, a sub-space of the TATT
model which includes a redshift dependence, is used to model intrinsic
alignments, resulting in a $\sim 1\sigma$ shift toward a higher value of
$S_{8}$.

Table 2: Mean posterior value of $S_{8}$ and 1$\sigma$ error. $\chi^{2}_{\rm min}$ gives the minimum values of $\chi^{2}$ for each analysis variant. Here, $\hat{\chi}^{2}_{\rm min}=\chi^{2}_{\rm min}/N_{\rm deg}$, where $N_{\rm deg}$ is the number of degrees of freedom. Following Paper I, for a number of data points $N_{\rm DP}$, we have $N_{\rm deg}=N_{\rm DP}-4.5$ for free fits and $N_{\rm deg}=N_{\rm DP}-2.5$ for Planck fits (Amon & Efstathiou, 2022). This counting of effective degrees of freedom is approximate but relatively unimportant since $N_{\rm DP}$ is much greater than unity. The column labelled $N_{\sigma}$ lists $(\chi^{2}-N_{\rm deg})/\sqrt{2N_{\rm deg}}$. The column labelled $N_{\rm FP}$ lists the number of free parameters in each model. The starred indexes report two results from DES22 (see their Table 3, entries 2 and 3). The first is the $\Lambda{\rm CDM}$ Optimised result, ie applying scale cuts (SC). The second uses the simpler NLA model for intrinsic alignments and fixes the neutrino mass. These two changes in analysis choices are maintained throughout Variants 3-10. Variant 3 (and subsequent variants) uses HMCode2020 to model the non-linear matter power spectrum without feedback. $A_{i}$ extensions sample the power spectrum in $5$ bins in wavenumber $k$ (Sec. 5). Variants 12-13 report the constraints on $S_{8}$ using data from KiDS. All other variants use DES Y3. Cases labelled ‘Planck ’ indicate those where the Planck prior is applied to the cosmological parameters, rather than uninformative priors, as in the ‘free’ case.  Variant | Data | Small-scale model | Other | Cosmology | $S_{8}$ | $N_{\rm DP}$ | $N_{\rm FP}$ | $\chi^{2}_{\rm min}$ | $N_{\sigma}$ | $\hat{\chi}^{2}_{\rm min}$
---|---|---|---|---|---|---|---|---|---|---
1* | DES | SC |  Halofit, TATT, free-$\nu$ | Free | $0.772^{+0.018}_{+0.017}$ | 273 | 14 | 285.0 | 1.14 | 1.06
2* | DES | SC | Halofit | Free | $0.788^{+0.017}_{+0.016}$ | 273 | 14 | 288.1 | 1.28 | 1.07
3 | DES | SC |  | Free | $0.805\pm{0.017}$ | 273 | 14 | 283.7 | 0.66 | 1.06
4 | DES | - |  | Free | $0.793\pm{0.012}$ | 400 | 14 | 418.1 | 0.80 | 1.06
5 | DES | $A_{\rm mod}$ |  | Free | $0.831\pm{0.036}$ | 400 | 15 | 417.1 | 0.77 | 1.05
6 | DES | $A_{\rm mod}$ |  | Planck | $0.811\pm{0.010}$ | 400 | 3 | 420.7 | 0.80 | 1.06
7 | DES |  $A_{\rm mod}$ \+ SC |  | Planck | $0.807\pm{0.012}$ | 273 | 3 | 289.4 | 0.81 | 1.07
8 | DES | $C_{\rm mod}(z)$ |  | Free | $0.842\pm{0.022}$ | 400 | 20 | 413.1 | 0.63 | 1.04
9 | DES | $C_{\rm mod}(z)$ |  | Planck | $0.813\pm{0.009}$ | 400 | 8 | 419.3 | 0.77 | 1.05
10 | DES | $A_{i}$ |  | Free | $0.810\pm{0.025}$ | 400 | 19 | 414.2 | 0.85 | 1.05
11 | DES | $A_{i}$ |  | Planck | $0.813\pm{0.009}$ | 400 | 7 | 418.1 | 0.73 | 1.05
12 | KiDS | $A_{\rm mod}$ |  | Free | $0.780\pm{0.035}$ | 225 | 21 | 260.3 | 1.89 | 1.18
13 | KiDS | $A_{\rm mod}$ |  | Planck | $0.808\pm{0.013}$ | 225 | 3 | 265.5 | 2.04 | 1.19

Variant 3 in Table 2 uses HMCode2020 dark matter model for the non-linear
power spectrum, instead of Halofit and simplifies the intrinsic alignment
model even further using the NLA model without a redshift-dependence (which
Secco, Samuroff et al. (2022) have shown gives an acceptable fit to the DES Y3
data). As in variant 2, the cosmological and nuisance parameters are allowed
to vary over the prior ranges given in Table 1. With this set of analysis
choices, the mean posterior value of $S_{8}$ in variant 3 is about $1\sigma$
higher than the value from variant 2. We refer the reader to DES & KiDS
Collaborations (2023) for a detailed analysis of the impact of each these
modelling strategies.

Variant 4 is the same as variant 3 except that it uses $\xi_{\pm}$
measurements over the entire angular ranges plotted in Fig. 1 333Note that
although the DES Y3 scale cuts were chosen to reduce the sensitivity of
$S_{8}$ to baryonic feedback, small angular scales are subject to other
systematics. The most important of these concerns is the modelling of the
point spread function, which has been shown to be unimportant on the angular
scales used in DES22. Further work is required to assess the impact of PSF
uncertainties on $\xi_{\pm}$ on the angular scales falling within the shaded
regions of Fig. 1.. The best-fit models from variants 3 and 4 are compared to
the DES Y3 measurements in Fig. 1 and although baryon feedback is not
accounted for in variant 4, they both provide excellent (almost
indistinguishable) fits to the data. The $\chi^{2}$ values for these fits are
given in Table 2.

Comparing the posterior widths of $S_{8}$ with the Planck  TTTEEE result of
$S_{8}=0.828\pm 0.016$ (Efstathiou & Gratton, 2021), we see that in variants 3
and 4, $S_{8}$ is lower by about $0.9\sigma$ and $1.8\sigma$, respectively.
One might therefore conclude that with the choices made in this paper (i.e.
adopting the NLA intrinsic alignment model and HMCode2020) the DES Y3 weak
lensing measurements are consistent with Planck  $\Lambda{\rm CDM}$ even if
baryonic feedback effects are ignored. This inference holds, apparently, even
for variant 4, which has a smaller error on $S_{8}$ and is sensitive to
spatial scales at which all cosmological hydrodynamical simulations show power
spectrum suppression caused by baryonic feedback. However, focusing on the
$S_{8}$ parameter gives a misleading impression of consistency because
cosmological parameters allowed by DES are disfavoured by Planck. As we will
show in the next section, suppression of the non-linear power spectrum is
required at high significance to reconcile the Planck $\Lambda{\rm CDM}$
cosmology with DES (and KiDS) weak lensing measurements.

##  3 Small-scale power suppression: constraints on the parameter $A_{\rm
mod}$

Following Paper I, in this section we incorporate the $A_{\rm mod}$ parameter
to model suppression of power on non-linear scales via Eq. 1. Variants 5 and 6
in Table 2 use the full range of angular scales plotted in Fig. 1. Variant 5
allows the cosmological parameters to vary freely over their priors, i.e. it
is identical to variant 4 but with the addition of the $A_{\rm mod}$
parameter444Aricò et al. (2023) report constraints using DES weak lensing
measurements without scale cuts (see also Chen et al., 2022a). Aricò et al.
(2023) use the baryonification prescription (Schneider & Teyssier, 2015; Aricò
et al., 2020) to model baryonic feedback but choose priors that restrict the
strength of the feedback. Variant 6 includes the Planck prior of Eq. 2.

Fig. 2 (left-hand side) compares the DES constraints from variants 5 (red) and
6 (yellow) in the $A_{\rm mod}-S_{8}$ plane. As shown in Paper I and in Fig.
2, the $A_{\rm mod}$ parameter is strongly degenerate with the linear theory
value of $S_{8}$. The vertical dashed line in Fig. 2 shows the best fit Planck
$\Lambda{\rm CDM}$ value $S_{8}=0.828$. The inclusion of the Planck prior
tightens the constraints significantly555Note that when we apply the $Planck$
prior, the nuisance parameters are fixed to their best-fit values. We have
verified that freeing the nuisance parameters does not significantly alter the
constraint on $A_{\rm mod}$. Similarly, the use of the $Planck$ prior does not
cause any significant shifts in the nuisance parameters compared to the DES Y3
fiducial analysis. These tests show that the value of $A_{\rm mod}$ in Eq. 3
is insensitive to the nuisance parameters.. One can see that adding the Planck
prior shifts the best fit value of $S_{8}$ downwards, but within about
$1\sigma$ of the best fit $\Lambda{\rm CDM}$ value measured by Planck  alone.
The Planck $\Lambda{\rm CDM}$ cosmology is therefore compatible with the DES
Y3 weak lensing measurements provided that the power spectrum is suppressed on
non-linear scales with

$A_{\rm mod}=0.858\pm{0.052},\qquad{\rm DES\ \ Y3\ (no\ scale\ cuts}).$ (3)

Thus if no scale cuts are applied, $A_{\rm mod}$ differs from unity by about
$2.7\sigma$. Note that the best fit model in variant 6 is plotted as the
orange line in Fig. 1 and is almost indistinguishable from the best fits to
variants 3 and 4 which allow cosmological parameters to vary freely.

Variant 7 is the same as variant 6 but applies the DES scale cuts. For this
case,

$A_{\rm mod}=0.919\pm{0.099},\qquad{\rm DES\ \ Y3\ (scale\ cuts}).$ (4)

The error bar increases compared to Eq. 3 and now the parameter $A_{\rm mod}$
differs from unity by only about $0.8\sigma$. This, of course, does not
conflict with Eq. 3 and tells us merely that large angular scales are not as
sensitive to power spectrum suppression as smaller angular scales.

The choice of scale cuts in the DES Y3 analysis was based on the baryonic
feedback effects measured from the EAGLE (McAlpine et al., 2016) and from the
OWLS-AGN (van Daalen et al., 2011) simulations, as detailed in Krause et al.
(2021); Secco, Samuroff et al. (2022). Using these simulations, scale cuts
were determined which resulted in a maximum two-dimensional bias in the
$\Omega_{\rm m}-S_{8}$ plane of $0.14\sigma_{2D}$ for the cosmic shear
analysis. DES22 reported cosmological constraints by applying these scale cuts
instead of modelling baryonic feedback effects. Our results in Eq. 4 suggest
that this strategy does largely reduce the sensitivity of cosmological results
to power suppression on non-linear scales, since the value of $S_{8}$ in
variant 3 is within $0.8\sigma$ of the Planck $\Lambda{\rm CDM}$ value.
However, as scale cuts are used to mitigate baryonic effects, with no attempt
to model feedback, the exact biases introduced into the $S_{8}$ parameter such
will depend on the accuracy of the OWLS-AGN simulation as an upper bound on
baryonic feedback effects. One cannot rule out small biases towards low values
of $S_{8}$ if the baryonic feedback is actually stronger than in these
simulations (or if the power spectrum suppression is caused by the properties
of the dark matter). The results in Eq. 4 perhaps hint that this might be the
case. What is clear, however, is that when the scale cuts are removed,
significant suppression of the non-linear spectrum is required to reconcile
the DES Y3 lensing results with the Planck $\Lambda{\rm CDM}$ cosmology.

The values of $A_{\rm mod}$ in Eqs. 3 and 4 are both higher than the value
reported in Paper I from an analysis of KiDS weak lensing. We compare KiDS
with the result of Eq. 3, where no scale cuts are used in either case (though
we note that KiDS measurements extend to smaller angular scales than DES). In
the analysis of KiDS reported in Paper I, to infer a value of $A_{\rm mod}$ we
kept the $\Lambda{\rm CDM}$ cosmological parameters fixed to the Planck  best
fit values, whereas in this paper we have applied the Planck  prior of Eq. 2.
Including a Planck  prior skews $A_{\rm mod}$ towards higher values since the
joint likelihood peaks at slightly lower values of $S_{8}$. We have repeated
the analysis of Paper Iusing $\xi_{\pm}$ including the Planck  prior (variant
13 in Table 2) finding

$A_{\rm mod}=0.748\pm{0.072},\qquad{\rm KiDS},$ (5)

which is $\sim 1\sigma$ higher than the value reported in AE. The joint
constraints on $A_{\rm mod}$ and $S_{8}$ from this variant are shown by the
filled blue contours in Fig. 2. The constraints from KiDS where cosmological
parameters are allowed to vary freely (variant 12) are shown by the blue
dotted contours. These constraints are similar to those from DES, though
displaced to lower values of $S_{8}$. The differences between KiDS and DES are
consistent with sampling fluctuations. Naively combining the two estimates
Eqs. 3-5 we find

$A_{\rm mod}=0.820\pm{0.042},\qquad{\rm DES\ \ Y3+KiDS}.$ (6)

Thus, according to our model, to reconcile the Planck base $\Lambda{\rm CDM}$
cosmology with DES+KiDS weak lensing data requires power spectrum suppression
on small scales at high statistical significance ($\sim 4\sigma$).

Figure 2: Left: Illustration of the strong degeneracy between $S_{8}$ and the
phenomenological power spectrum suppression parameter $A_{\rm mod}$ and the
inferred power suppression on non-linear scales. Left: the 68% and 95%
constraints for DES $\xi_{\pm}$ statistics (without angular scale cuts)
analyzed using HMCode2020-no feedback, varying $A_{\rm mod}$ with free
cosmology (red) and similarly for KiDS $\xi_{\pm}$ (blue), as presented in
Paper I. The dashed line indicates the Planck $\Lambda{\rm CDM}$ best-fit
value of $S_{8}=0.828$. The DES constraint when incorporating the Planck-prior
is shown in orange, and the KiDS equivalent in green. Right: the resulting
power suppression corresponding to the DES/KiDS Planck-prior $A_{\rm mod}$
posteriors (orange/green). The DES data require less suppression of power
compared to KiDS to be consistent with the Planck LCDM cosmology. As in Paper
I, we compare to predictions from various hydrodynamical simulations (black).

The power spectrum suppression corresponding to Eqs. 3 and 5 at $z=0$ is shown
in the right-hand panel of Fig. 2. In this and subsequent figures, we compare
our predictions to the power spectrum suppression measured from hydrodynamical
simulations taken from van Daalen et al. (2020). The solid and dotted lines
correspond to the lower and upper bounds on the power spectrum suppression
suggested by the BAHAMAS simulations (McCarthy et al., 2017, BAryons and
HAloes of MAssive Systems). These correspond to their AGN feedback parameter
of $\log_{10}(\Delta T_{\rm heat}/K)=7.6$ and $8.0$ respectively. With the
feedback prescription adopted in BAHAMAS, the choice $\log_{10}(\Delta T_{\rm
heat}/K)=7.8$ provides a good match to the observed gas fractions in groups
and clusters and to the stellar mass function at $z=0.1$ (van Daalen et al.,
2020). The feedback prescription used in the two C-OWLS simulations (Le Brun
et al., 2014, COSMO-Overwhelmingly Large Simulations) is the same as in
BAHAMAS and uses higher values of log${}_{10}(\Delta T_{\rm heat}/K$). As a
result, the power spectrum suppression in C-OWLS is stronger and extends to
lower wavenumbers than in the BAHAMAS simulations. As discussed in van Daalen
et al. (2020) the stronger feedback in these simulations resulted in deficit
of galaxies with stellar masses below $10^{11}h^{-1}M_{\odot}$ compared to
observations. For reference, the power spectrum suppression in the OWLS-AGN
simulation used to select the DES angular scale cuts is similar to that in the
fiducial BAHAMAS simulations with $\log_{10}(\Delta T_{\rm heat}/K)=7.8$. The
suppression in the EAGLE simulations is much weaker than any of the
simulations plotted in Fig. 2, see Chisari et al. (2019).

To reconcile Planck $\Lambda{\rm CDM}$ with weak lensing based on DES and
KiDS, the power spectrum on non-linear scales must be more strongly suppressed
than in the BAHAMAS simulation with $\log_{10}(\Delta T_{\rm heat}/K)=8$
(which over the wavenumber range $k\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}1h{\rm Mpc}^{-1}$ is well
approximated by $A_{\rm mod}=0.9$). Stronger power spectrum suppression,
closer to the two C-OWLS models plotted in Fig. 2 is required to match the
results of Eqs. 3 and 5.

The one parameter $A_{\rm mod}$ model has the virtue of simplicity, but
severely restricts the functional form of any power suppression. Ideally, one
would want to reconstruct both the redshift and wavenumber dependence of the
matter power spectrum directly from the observations. However, current weak
lensing data are not sufficiently powerful to allow a full 2-dimensional
reconstruction. The next two sections should therefore be considered
exploratory. Sec. 4 investigates a simple parametric form for the redshift
dependence, while Sec. 5 investigates the wavenumber dependence of the power
spectrum by splitting $A_{\rm mod}$ into a coarse set of bins in $k$.

##  4 Redshift dependent power suppression

Figure 3: Trajectories of $C_{\rm mod}(z)$ with $\Delta\chi^{2}<1$ over the
redshift range $z=0-1.5$. Overlaid black data points represent the weighted
means of the trajectories, evaluated in intervals of $\Delta z=0.2$. The
errors plotted here are at the 95% confidence intervals and are asymmetric
because the distributions are non-Gaussian (as can be seen by the outliers).
The trajectories are constrained most accurately in the redshift region of
$z\approx 0.3$; the redshift at which weak lensing surveys are most sensitive.
At lower and higher redshifts the trajetories are not well constrained and
depend sensitively on the choices of priors. The colour gradient indicates the
corresponding value $C_{\rm mod}(z=0)=A$ with the parameterisation of Eq. 8.
Figure 4:  Power suppression predicted by the $C_{\rm mod}(z)$ model at a
redshift $z=0.25$ (approximately the redshift at which DES weak lensing gives
the tightest constraints) compared to the suppression seen in numerical
simulations at this redshift.

As discussed in DES22, the DES Y3 shear analysis is performed using four
tomographic redshift bins with mean redshifts of $0.34$, $0.52$, $0.74$ and
$0.96$. The mean redshift of the entire sample is $z_{\rm mean}\sim 0.63$ so
we would expect the weak lensing statistics to be most sensitive to the matter
distribution at a redshift $z\sim z_{\rm mean}/2\sim 0.3$ and to have little
sensitivity to the matter distribution at higher redshifts.

This reasoning can be made more quantitative by analysing a parametric form of
the redshift dependence of the matter power suppression. We chose the
following functional form

$C_{\rm mod}(z)=A[1-f_{1}(z)]+B[f_{2}(z)],$ (7a) where the functions
$f_{k}(z)$ are $f_{k}(z)=\frac{(z/z_{\rm c_{k}})^{\alpha_{k}}}{1+(z/z_{\rm
c_{k}})^{\alpha_{k}}},\hskip 5.69054ptk=1,2.$ (7b)

The matter power spectrum suppression according to this model is

$P_{\rm m}(k,z)=P^{\rm L}_{\rm m}(k,z)+C_{\rm mod}(z)[P^{\rm NL}_{\rm
m}(k,z)-P^{\rm L}_{\rm m}(k,z)],$ (8)

and uses six parameters $\\{A$, $B$, $z_{c_{1}}$, $\alpha_{1}$, $z_{c_{2}}$,
$\alpha_{2}\\}$ instead of the single parameter $A_{\rm mod}$ of Eq. 1. We
adopt flat priors on these parameters as summarized in Table 3.

Table 3: Parameters of the $C_{\rm mod}(z)$ model of Sec. 4, prior ranges and their marginal mean values in both free and Planck prior base $\Lambda{\rm CDM}$ cosmologies. The priors on the parameters are uniform over the ranges given in the square brackets. Note that the redshift reconstructions are sensitive to the priors as noted in the text. Parameter | Prior | Marginal Mean |
---|---|---|---
|  | free cosmology |  Planck  prior |
$A$ | F[0.5, 1.0] | $0.893\pm{0.074}$ | $0.914\pm{0.055}$ |
$B$ | F[0.5, 1.0] | $0.759\pm{0.132}$ | $0.757\pm{0.142}$ |
$z_{c_{1}}$ | F[0.1, 1.0] | $0.674\pm{0.149}$ | $0.768\pm{0.137}$ |
$z_{c_{2}}$ | F[1.0, 4.0] | $2.511\pm{0.817}$ | $2.472\pm{0.838}$ |
$\alpha_{1}$ | F[1.0, 4.0] | $2.472\pm{0.749}$ | $2.711\pm{0.728}$ |
$\alpha_{2}$ | F[1.0, 4.0] | $2.491\pm{0.793}$ | $2.435\pm{0.860}$ |

Figure 4 compares the power spectrum suppression constraints from variant 9
(purple contours) with those from the one-parameter $A_{\rm mod}$ model of
variant 6 (orange contours). We show the results at a redshift of $z=0.25$,
since at this redshift the results are insensitive to our choice of the
fitting function and priors. The purple contours span a slightly broader range
than the orange contours, since in the $C_{\rm mod}(z)$ model, one can trade
off increased suppression at $z\sim 0.3$ with decreases in suppression at
higher and lower redshifts and vice-versa. We also show the suppression
measured in the cosmological hydrodynamics simulations at $z=0.25$. These are
almost identical to the simulation results at $z=0$ shown in Fig. 2.

Figure 5: The best fit prediction for $\xi_{\pm}(\theta)$ (black) and the
contributions of different $k$ ranges to the overall best fit are given in
different colours, highlighting the relative power of each $k$-bin. The
optimised $\Lambda{\rm CDM}$ scale cuts from the DES Y3 analysis (Amon et al.,
2023); (Secco, Samuroff et al., 2022) are in shaded blue. The largest spatial
scales (blue), corresponding to $k<0.1$, contribute to $\xi_{+}$ at large
$\theta$ but make a very small contribution to $\xi_{-}$. The smallest scales
(pink), $k>10^{0.5}$, contribute to $\xi_{-}$ on small angular scales but make
a very small contribution to $\xi_{+}$.

In conclusion, the results of this section show that the DES data lack the
statistical power to reconstruct the redshift dependence of the power
suppression. Most of the statistical power of DES weak lensing comes from the
matter distribution at $z\sim 0.3$, with relatively little sensitivity to the
behaviour at higher or lower redshifts. The one parameter $A_{\rm mod}$ model
cannot therefore be extrapolated beyond a relatively narrow range of redshifts
centred at $z\sim 0.3$.

##  5 Scale-dependent suppression

Figure 6: The blue bands show the $1\sigma$ ranges of the reconstruction of
the power suppression using 5 wavenumber bins $A_{i}$ as in Eq. 9. The
constraints on the $A_{\rm mod}$ parameterisation from Fig. 2 are shown in
orange. Note that the $A_{i}$ coefficients are strongly correlated with each
other. We also plot the suppression measured in numerical hydrodynamicals
simulations as in Fig. 2.

In this section, we explore the scale-dependence of the non-linear matter
power spectrum suppression by replacing the $A_{\rm mod}$ parameter with five
parameters, $A_{i}$m separated in wavenumber $k$ as follows:

$\left.\begin{array}[]{ll}A_{1}:&\ \ \ \ \ \ \ \ \ \ \log_{10}k\leq-1,\\\
A_{2}:&\ \ -1<\log_{10}k\leq 0.5,\\\ A_{3}:&-0.5<\log_{10}k\leq 0,\\\ A_{4}:&\
\ \ \ 0<\log_{10}k\leq 0.5,\\\ A_{5}:&\ \ \ \ \ \ \ \ \ \
\log_{10}k>0.5.\end{array}\right\\}$ (9)

where $k$ is in units of $h{\rm Mpc}^{-1}$ throughout this section.

To begin, we examine the contribution of each of the $k$-bins to the
$\xi_{\pm}(\theta)$ prediction of the DES Y3 data666Specifically, we use the
best-fit parameters from the analysis variant 4 in Table 2, with no scale
cuts, shown in Fig. 5. We make several observations: (i) as is well known at
any given angular scale $\theta$ the $\xi_{\pm}$ statistics mix of information
from a wide range of $k$-scales; (ii) $\xi_{-}$ is significantly more
sensitive to non-linear information compared to $\xi_{+}$; (iii) the bin
covering linear scales ($k<0.1$, plotted in blue) makes little contribution to
$\xi_{-}$ over the measured angular range, though it is important for the
larger $\theta$ values of $\xi_{+}$; (iv) the DES scale cuts effectively
remove sensitivity to wavenumbers $k>1$, thus small angular scales that are
within the scale cuts must be included to constrain the parameters $A_{4}$ and
$A_{5}$.

Table 4: Parameters of the binned $A_{\rm mod}$ model. Prior ranges and marginal mean values in both free and fiducial Planck base $\Lambda{\rm CDM}$ cosmologies are given. Prior ranges are broad to allow for all possible trajectories to be investigated and are uniform (denoted by square brackets). Parameter | Prior | Marginal Mean |
---|---|---|---
|  | free cosmology |  Planck prior |
$A_{1}$ | F[0.5, 1.0] | $0.838\pm{0.190}$ | $0.854\pm{0.200}$ |
$A_{2}$ | F[0.5, 1.0] | $0.913\pm{0.181}$ | $0.815\pm{0.195}$ |
$A_{3}$ | F[0.5, 1.0] | $0.945\pm{0.138}$ | $0.931\pm{0.117}$ |
$A_{4}$ | F[0.5, 1.0] | $0.894\pm{0.106}$ | $0.903\pm{0.075}$ |
$A_{5}$ | F[0.5, 1.0] | $0.757\pm{0.117}$ | $0.741\pm{0.104}$ |

We adopt flat priors on each $A_{i}$ parameter, as summarized in Table 4, with
the lower limits describing much more extreme power suppression than seen in
cosmological hydrodynamical simulations. The posterior mean value of $S_{8}$
and reduced-$\chi^{2}$ for the $A_{i}$ analysis are given in Table 2 for both
the free cosmology and Planck prior cases (variants 10 and 11 respectively).
The mean value of the posterior and the standard deviations of the $A_{i}$
parameters are reported in Table 4.

Fig. 6 shows the predicted power suppression in each $k$-bin (separated by
grey vertical lines), compared to the fiducial $A_{\rm mod}$ for the fits that
include the Planck prior. The constraints from the binned model track the
general shape and amplitude of the one-parameter $A_{\rm mod}$ model. The main
new result from this analysis is that power suppression of $\sim 3-10$%
spanning mildly non-linear scales (bin 2, spanning wavenumbers in the range
$0.1<k<0.5$) is required to reconcile the Planck$\Lambda{\rm CDM}$ data with
the DES weak lensing data. It is not possible to avoid suppression in bin 2 by
increasing the suppression at smaller scales, mainly because $\xi_{-}$ is
dominated by bin 2 over the angular range $\theta\sim
40^{\prime}-100^{\prime}$ (see the green curves in Fig. 5).

Fig. 6 also shows the power spectrum suppression measured in the BAHAMAS and
C-OWLS simulations (as in Fig. 2). Evidently, if baryonic feedback is
responsible for the apparent $S_{8}$ tension, the analysis of this section
shows that the feedback must propagate to scales $k\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.3$. This requires stronger
feedback than in the BAHAMAS simulation with $\log_{10}(\Delta T_{\rm
AGN}/{\rm K})=7.8$ favoured by (McCarthy et al., 2017), in agreement with the
conclusions of Sec. 3.

##  6 Discussion and Conclusion

Figure 7: The Planck TTTEEE, both with and without Planck lensing (Efstathiou
& Gratton, 2021) and the ACT+BAO constrains (Madhavacheril et al., 2023)
$1\sigma$ constraints on $S_{8}$ are shown in grey. Below these, in black,
blue and green, are the results for the KiDS data analysed with no scale cuts,
no scale cuts with $A_{\rm mod}$ (from AAGE) and no scale cuts with both a
Planck prior and $A_{\rm mod}$ variants 12 and 13 from Table 2 respectively.
The next three results show in order the results of variants 4, 3 and adding a
Planck prior to the DES analysis of variant 3, respectively (black). Finally,
the results of variants 5 and 6 are shown in red and yellow respectively. This
illustrates the sensitivity of the $S_{8}$ tension on scale cuts and the
modelling of non-linear scales. Figure 8: A rough guide to the approximate
scale-dependence in terms of wavenumber, $k$, and redshift-dependence, $z$, of
cosmological observations. CMB lensing measurements are consistent with Planck
$\Lambda{\rm CDM}$ (both blue, filled) and have negligible sensitivity on non-
linear modelling and span the range $z\sim 0.5-5$. Weak galaxy lensing is
sensitive to a wide range of scales at $z<1$, but primarily probes the non-
linear regime (red, filled). With future lensing data, is it possible to
separate the linear information from weak-lensing. Both redshift space
distortions and cross-correlations of CMB lensing with galaxy positions
typically limit their analyses to linear scales and are sensitive to lower
redshifts than CMB lensing. These two probes therefore provide a powerful test
of the non-linear solution to the $S_{8}$ tension proposed here. Lyman-alpha
measurements are also sensitive to a wide range of scales, but at higher
redshifts.

The aim of this investigation has been to assess whether the $S_{8}$ tension
can be resolved, that is Planck $\Lambda{\rm CDM}$ cosmology can be made
consistent with weak lensing observations by modifying the matter power
spectrum on non-linear scales. Following Paper I, we have investigated
constraints on the power suppression parameter, $A_{\rm mod}$ of Eq. 1, using
DES Y3 cosmic shear data. In this analysis we include a Planck  prior
describing their constraints on key cosmological parameters and the associated
uncertainties.

The DES data require substantial suppression of the matter power spectrum on
non-linear scales to become consistent with Planck. The suppression required
is less extreme than found from the KiDS weak lensing measurements, though the
results from these two surveys are statistically consistent. However, if such
a suppression is interpreted in terms of baryonic feedback, then it must be
stronger than the most extreme feedback prescription implemented in the
BAHAMAS simulations.

The constraints on $A_{\rm mod}$ depend on the angular scale cuts applied to
the $\xi_{\pm}$ measurements. If the DES ‘$\Lambda{\rm CDM}$-Optimised’
angular scale cuts are imposed on $\xi_{\pm}$, the cosmological constraints
from DES data are degraded and are statistically compatible with the Planck
cosmology. For this case, $A_{\rm mod}$ is consistent with unity, though with
a large error.

We have analysed the DES Y3 data using an extended $A_{\rm mod}$ model that
includes either a redshift or wavenumber dependence. The DES data have little
sensitivity to redshifts outside of a relatively narrow range centred at
$z\sim 0.3$. The one parameter $A_{\rm mod}$ model, therefore, provides an
adequate approximation at this redshift but cannot be extrapolated reliably to
higher or lower redshifts.

To investigate the wavenumber dependence, we solved for amplitude suppression
factors $A_{i}$ in five logarithmically spaced bins. The results show that
consistency between DES and Planck $\Lambda{\rm CDM}$ requires suppression on
scales $k\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.3\ h/\rm{Mpc}$.
This result is in agreement with our results for $A_{\rm mod}$ and shows that
the requirement of the data for power suppression on these scales is not an
artefact of the simple $A_{\rm mod}$ parameterisation.

Fig. 7 summarizes both the updated results of Paper I and this paper. The
entry labelled Planck TTTEEE shows the Planck base $\Lambda{\rm CDM}$
constraints from Efstathiou & Gratton (2021) and the entry labelled Planck
TTTEEE+lensing includes the Planck  lensing likelihood (Planck Collaboration
et al., 2020b). ACT lensing+BAO shows the $\Lambda{\rm CDM}$ constraints from
the recent ACT CMB lensing results combined with baryon acoustic oscillation
measurements (Madhavacheril et al., 2023). These measurements are derived from
predominantly linear scales ($k\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.1\ h/\rm{Mpc}$) and demonstrate
that the $\Lambda{\rm CDM}$ cosmology provides a consistent description of the
matter fluctuations from the redshift of recombination to the low redshifts
that dominate the CMB lensing signal, $z\sim 0.5-5$.

To compare the results from the CMB with those from weak gravitational
lensing, it is necessary to extract the linear amplitude of the matter
fluctuations from statistics that are dominated by non-linear scales. This
requires an accurate model of the dark matter power spectrum on non-linear
scales, including modifications caused by baryonic feedback. Incorrect
modelling of the non-linear spectrum can therefore lead to an apparent tension
between weak lensing estimates of $S_{8}$ and those measured from the CMB.

This is illustrated by the remaining entries in Fig. 7 which summarize results
from the KiDS and DES Y3 weak lensing. One can see that there are varying
degrees of tension with the CMB. The two entries labelled ‘KiDS $\theta>0.5\
{\rm arcmin}$’ (from Paper I) and ‘DES $\theta>2.5\ {\rm arcmin}$’ (variant 4)
use the full angular ranges of $\xi_{\pm}$ reported by the two surveys (i.e.
no scale cuts). For these entries Fig. 7 shows results on $S_{8}$ allowing
cosmological parameters to vary freely, using HMCode2020 to model the non-
linear spectrum and ignoring baryonic feedback. The ‘tension’777The numbers
quoted here are based on the simple expression
$(S^{1}_{8}-S^{2}_{8})/\sqrt{\sigma^{2}_{S^{1}_{8}}+\sigma^{2}_{S^{2}_{8}}}$.
with the Planck  TTTEEE+lensing entry is $\sim 3.7\sigma$ for KiDS and $\sim
2.1\sigma$ for DES. If we apply the DES $\Lambda{\rm CDM}$ -Optimised scale
cuts (variant 3) the tension with the Planck $S_{8}$ drops to $\sim 1\sigma$.
This suggests that the scale cuts have largely eliminated sensitivity to non-
linear power spectrum suppression, at the expense of increasing the error on
$S_{8}$, leading to consistency with Planck. The fact that DES $S_{8}$ in this
case is slightly low may be because the effects of non-linear power spectrum
suppression have not been eliminated entirely.

If we impose a Planck  prior and include the parameter $A_{\rm mod}$, the DES
and KiDS lensing can be made compatible with Planck, with or without scale
cuts. The question then is whether the small scale suppression required is
physically reasonable. As summarized above, if it is caused by baryonic
feedback, then the suppression must be stronger than favoured in recent
cosmological hydrodynamic simulations. However, underestimating baryonic
effects is not the only plausible interpretation since the suppression may
reflect non-standard properties of the dark matter on non-linear scales (see
e.g. Poulin et al., 2022; Rogers et al., 2023).

If the weak-lensing $S_{8}$ tension is caused by suppression of power on non-
linear scales, what would we expect for other cosmological measurements? Fig.
8 shows a sketch of the wavenumber and redshift ranges covered by various
cosmological observations. As noted in Paper I, all measures of $S_{8}$ at low
redshift from linear scales should agree with the Planck cosmology.
Specifically, CMB lensing measurements combined with baryon acoustic
oscillations, such as recent measurements from Advanced ACT are consistent
with the Planck cosmology (Madhavacheril et al., 2023). Cross-correlations of
CMB lensing with galaxy surveys can provide an important test of this
hypothesis. Forthcoming analyses using the latest CMB lensing maps from ACT
(Qu et al., 2023) and the South Pole Telescope (Omori et al., 2023) should
agree with the predictions of the Planck $\Lambda{\rm CDM}$ model.

As reviewed in Paper I redshift-space distortions measured from linear scales
from current galaxy redshift surveys are consistent with our interpretation,
but not to high precision. This situation should change decisively in the near
future with RSD measurements from DESI (see e.g. Schlegel et al., 2022, and
references therein). Measurements of $S_{8}$ from quasar Ly-alpha lines from
DESI uniquely give access to high $k$ and $z\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}1$ and offer an opportunity to
potentially break the degeneracy between baryonic feedback and non-standard
dark matter. Tests of a redshift-dependent growth, $\sigma_{8}(z)$ or
$S_{8}(z)$ using measurements from from linear scales should be consistent
with $\Lambda{\rm CDM}$ (White et al., 2022; DES Collaboration et al., 2022;
García-García et al., 2021)888However, deviations from the $\Lambda{\rm CDM}$
growth rate have been reported when linear measurements are combined with weak
galaxy lensing measurements that are sensitive to non-linear scales (García-
García et al., 2021; Abbott et al., 2023)..

Weak galaxy lensing measurements offer a window to a wide range of scales as
indicated in Fig. 8. In the future, it may be possible to reconstruct the
power spectrum as a function of wavenumber and redshift from weak lensing
data. Assuming that systematic errors can be sufficiently mitigated, it may
become possible to accurately test the $\Lambda{\rm CDM}$ cosmology on linear
scales and to reconstruct the matter power spectrum on small scales using weak
galaxy lensing data alone. Such an analysis would establish unambiguously
whether the $S_{8}$ tension is caused by a deviation from the $\Lambda{\rm
CDM}$ model at late times, or whether it is an apparent tension caused by
physics on non-linear scales.

## Acknowledgements

The authors would like to thank Joe Zuntz and Niall MacCrann for help in
handling cosmosis (Zuntz et al., 2015). CP would like to thank Jessie Muir and
Sujeong Lee for helpful discussions on modified growth of structure, and
Anthony Challinor for useful discussions on CMB lensing. Alexandra Amon is
supported by a Kavli Fellowship. George Efstathiou is supported by an
Leverhulme Trust Emeritus Fellowship. Calvin Preston is supported by a Science
and Technology Facilities Council studentship.

This project has used public archival data from both the Dark Energy Survey
and the Kilo Degree Survey. 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
FacilitiesCouncil 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. Based on
observations made with ESO Telescopes at the La Silla Paranal Observatory
under programme IDs 177.A-3016, 177.A-3017, 177.A-3018 and 179.A-2004, and on
data products produced by the KiDS consortium. The KiDS production team
acknowledges support from: Deutsche Forschungsgemeinschaft, ERC, NOVA and
NWO-M grants; Target; the University of Padova, and the University Federico II
(Naples).

## Data availability

No new data were generated or analysed in support of this research.

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##  Appendix A Planck $\Lambda{\rm CDM}$ prior

Figure 9:  The degeneracies of $\Omega_{\rm m}$, $H_{0}$ and $S_{8}$ for the
base $\Lambda{\rm CDM}$ model using the Planck TTTEEE likelihood computed by
Efstathiou & Gratton (2021). The ellipses show the 68% and 95% contours of
$S_{8}$ and $\Omega_{m}$ (upper figure) and $S_{8}$ and $H_{0}$ (lower figure)
computed from the two dimensional Gaussian model of Eq. 2. The contours in the
lower panel assume Eq. 2 and a delta function $\delta(\Omega_{\rm
m}h^{3}-0.09612)$. The parameter combination $\Omega_{\rm m}h^{3}$ is an
approximation to the acoustic peak location parameter and is extremely well
determined by Planck  ($\Omega_{\rm m}h^{3}=0.09612\pm 0.00029$). The points
show samples from the COSMOMC chains.

In this appendix we justify the Planck  prior of Eq. 2. In the base
$\Lambda{\rm CDM}$ model, the spectral index $n_{\rm s}$ is well determined by
Planck and can be fixed to the Planck best fit value. The main parameters that
affect weak lensing are $S_{8}$, $\Omega_{\rm m}$ and $H_{0}$ (or combinations
of these parameters). The parameter combination $\Omega_{\rm m}h^{3}$ is a
proxy for the acoustic peak location parameter, $\theta_{*}$, and is extremely
well determined by Planck. The Planck $\Lambda{\rm CDM}$ degeneracies in the
space of $S_{8}$, $\Omega_{\rm m}$ and $H_{0}$ can be approximated accurately
by adopting a two dimensional Gaussian in $S_{8}$ and $\Omega_{\rm m}$ and
imposing a delta function constraint on the parameter combination $\Omega_{\rm
m}h^{3}$. The specific form of Eq. 2 comes from the inverse of the covariance
matrix of the parameters $S_{8}$ and $\Omega_{\rm m}$ determined from the
COSMOMC chains. The degeneracies in the $S_{8}-\Omega_{\rm m}$ and
$S_{8}-H_{0}$ planes inferred from Eq. 2 are plotted in Fig. 9 and are
compared to samples from the COSMOMC chains.