# A non-linear solution to the $S_{8}$ tension?

Alexandra Amon1, George Efstathiou1
${1}$ Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3
OHA E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Weak galaxy lensing surveys have consistently reported a lower amplitude for
the matter fluctuation spectrum, as measured by the $S_{8}$ parameter, than
expected in the $\Lambda{\rm CDM}$ cosmology favoured by Planck. However, the
expansion history follows the predictions of the Planck $\Lambda{\rm CDM}$
cosmology to high accuracy, as do measurements of lensing of the cosmic
microwave background anisotropies. Redshift space distortion measurements also
appear to be consistent with Planck $\Lambda{\rm CDM}$. In this paper, we
argue that these observations can be reconciled with the Planck $\Lambda{\rm
CDM}$ cosmology if the matter power spectrum is suppressed more strongly on
non-linear scales than assumed in analyses of weak galaxy lensing. We
demonstrate this point by fitting a one-parameter model, characterising a
suppression of the non-linear power spectrum, to the KiDS-1000 weak lensing
measurements. Such a suppression could be attributed to new properties of the
dark matter that affect non-linear scales, or to a response of the matter
fluctuations to baryonic feedback processes that are stronger than expected
from recent cosmological simulations. Our proposed explanation can be tested
using measurements of the amplitude of the matter fluctuation spectrum on
linear scales, in particular via high precision redshift space distortion
measurements from forthcoming galaxy and quasar redshift surveys.

###### keywords:

cosmology: cosmological parameters, weak lensing, observations

##  1 Introduction

The standard $\Lambda{\rm CDM}$ cosmological model provides a remarkably good
fit to a number of observations, including anisotropies of the cosmic
microwave background (CMB; e.g. Bennett et al., 2013; Planck Collaboration et
al., 2020a; Aiola et al., 2020; Dutcher et al., 2021), baryon acoustic
oscillations (BAO; e.g. Alam et al., 2017; Blomqvist et al., 2019; de Sainte
Agathe et al., 2019; Alam et al., 2021) and the magnitude-redshift relation of
Type Ia supernovae (e.g. Betoule et al., 2014; Scolnic et al., 2017; Brout et
al., 2021). Despite these successes, there is some evidence of ‘tensions’ with
other astrophysical data at varying degrees of statistical significance.

At the level of the background expansion history, some late time measurements
of the Hubble constant disagree with the $\Lambda{\rm CDM}$ value inferred
from the CMB (e.g. Riess et al., 2016; Riess et al., 2019, 2021; Wong et al.,
2020) though others do not (Freedman et al., 2019, 2020). This problem has
become known as the ‘Hubble tension’ (for recent reviews see Freedman, 2021;
Shah et al., 2021; Abdalla et al., 2022).

At the level of perturbations, weak lensing surveys have reported measurements
of the amplitude of the fluctuation spectrum (e.g. Heymans et al., 2013;
Hikage et al., 2019; Hamana et al., 2020; Asgari et al., 2021; Amon et al.,
2022b; Secco et al., 2022). Specifically, cosmic shear surveys tightly
constrain the parameter combination111Where $\Omega_{\rm m}$ is the present
day matter density in units of the critical density, $\sigma_{8}$ is the root
mean square linear amplitude of the matter fluctuation spectrum in spheres of
radius $8h^{-1}{\rm Mpc}$ extrapolated to the present day, and $h$ is the
value of the Hubble constant $H_{0}$ in units of $100\ {\rm km}{\rm
s}^{-1}{\rm Mpc}^{-1}$. $S_{8}=\sigma_{8}(\Omega_{\rm m}/0.3)^{0.5}$,
consistently finding values that are lower than that expected according to the
Planck best fit $\Lambda{\rm CDM}$ cosmology. This discrepancy has become
known as the ‘$S_{8}$ tension’. Most recently, two large cosmic shear surveys
have reported new constraints. Assuming a spatially flat $\Lambda{\rm CDM}$
cosmology, the Kilo-Degree Survey (KiDS)-1000 gives222We quote results derived
from the COSEBI (complete orthogonal sets of E/B integrals, see Schneider et
al. (2010)) statistics.:

$S_{8}=\left\\{\begin{array}[]{ll}0.759^{+0.024}_{-0.021},&\mbox{KiDS--1000
cosmic shear},\\\ 0.766^{+0.020}_{-0.014},&\mbox{KiDS--1000 {3$\times$2}pt,
}\end{array}\right.$ (1)

see Asgari et al. (2021) for the shear-shear analysis (hereafter KiDS21) and
Heymans et al. (2021) for a $3\times 2{\rm pt}$ analysis combining shear-
shear, galaxy-galaxy lensing and galaxy-galaxy two-point statistics. The Dark
Energy Survey (DES) Year 3 analysis gives

$S_{8}=\left\\{\begin{array}[]{ll}0.772^{+0.018}_{-0.017},&\mbox{DES Y3 cosmic
shear},\\\ 0.779^{+0.014}_{-0.015},&\mbox{DES Y3 {3$\times$2}pt
}\end{array}\right.$ (2)

where we have quoted the ‘$\Lambda{\rm CDM}$ optimised’ results for the shear-
shear analysis (Amon et al., 2022b; Secco et al., 2022), hereafter DES22, and
for the 3x2pt function analysis (DES Collaboration et al., 2021). The results
quoted above are derived using different analysis choices (angular scale cuts,
intrinsic alignment model, etc.) and differences in assumptions concerning
neutrino masses333The Planck and KiDS analyses assume a normal hierarchy with
the heaviest neutrino mass fixed to $0.06\ {\rm eV}$, while DES allows the
neutrino mass to vary, which has little impact on their $S_{8}$ constraints
(DES Collaboration et al., 2021, Fig. 28)..

For Planck, we adopt the $\Lambda{\rm CDM}$ parameters reported in Efstathiou
& Gratton (2021) (hereafter EG21):

$\displaystyle S_{8}$ $\displaystyle=$ $\displaystyle 0.828\pm
0.016,\qquad{\rm TTTEEE},$ (3a) $\displaystyle S_{8}$ $\displaystyle=$
$\displaystyle 0.829\pm 0.012,\qquad{\rm TTTEEE+Plens},$ (3b)

where TTTEEE denotes the high multipole likelihood constructed by combining
the temperature power spectra (TT), temperature-polarization E-mode cross-
spectra (TE) and polarization E-mode power spectra (EE). Each of these
likelihoods is combined with low multipole ($\ell\leq 29$ ) TT and EE
likelihoods described in Planck Collaboration et al. (2020a). ‘Plens’ in Eq.
(3b) denotes the addition of the Planck CMB lensing likelihood (Planck
Collaboration et al., 2020b).

The KiDS-1000 cosmic shear measurements of $S_{8}$ are about 9% lower than the
Planck value of Eq. (3b), suggesting a discrepancy at the $\sim 2.4-2.7\sigma$
level444We quote $\Delta S_{8}/\sqrt{(\sigma^{\rm
Planck}_{S_{8}})^{2}+(\sigma_{S_{8}}^{\rm lensing})^{2}}$. depending on which
of the KiDS-1000 and Planck measurements are used in the comparison. This is
consistent with the conclusions of the KiDS team based on more complex tension
metrics (Asgari et al., 2021; Heymans et al., 2021). The DES Y3 measurements
are about $7\%$ lower than the Planck value, suggesting a discrepancy at about
the $\sim 2.3-2.6\sigma$ level555The DES team exclude Planck lensing when
comparing their results with Planck. As can be seen from Eqs (3a)-(3b) adding
Planck lensing reduces the error on $S_{8}$, increasing the significance of
the discrepancy with the DES Y3 results.. To simplify the analysis in this
paper, we focus on the cosmic shear measurement since it is the dominant
contribution in the $3\times 2{\rm pt}$ $S_{8}$ constraint and therefore the
driver of the $S_{8}$ tension.

While neither of the lensing surveys taken in isolation offers decisive
evidence for a discrepancy with the Planck $\Lambda{\rm CDM}$ cosmology, both
surveys find low values of $S_{8}$ in agreement with earlier work. It
therefore seems unlikely that the $S_{8}$ tension is simply a statistical
fluctuation. However, we note that it is naive to crudely combine the shear
estimates in Eqs (1) and (2), as these analyses use different modelling
frameworks, cosmological priors, angular ranges and various other analysis
choices such as to the modelling of intrinsic alignments (IA)666A combined
shear-shear analysis of DES Y3 and KiDS-1000, including a detailed analysis of
the differences in methodology and analysis choices, is currently in
preparation by the KiDS and DES collaborations..

In the last few years there have been significant advances in the calibration
of the lensing data (see Asgari et al., 2020; Amon et al., 2022b, and
references therein). Improved methods of calibrating photometric redshifts
(Hildebrandt et al., 2021; Myles et al., 2021) show that errors in the
redshift distributions of the source galaxies are unlikely to account for a
$\sim 7\%-9\%$ discrepancy in value of $S_{8}$. It also seems implausible that
systematic errors in the shear measurements could explain a discrepancy of
this size (Mandelbaum et al., 2018; Kannawadi et al., 2019; MacCrann et al.,
2022). Similarly, the modelling of IA may affect the value of $S_{8}$ at the
one or two percent level (given the small amplitudes of the alignment
corrections found by KiDS and DES) but it seems unlikely that IA are
responsible for a $\sim 7\%$ discrepancy.

What is the most likely explanation of this $S_{8}$ tension? Does it require a
radical departure from the $\Lambda{\rm CDM}$ paradigm? We explore both of
these questions in this paper. This paper is structured as follows. Section 2
discuss some preliminaries including comparisons with other types of data
which serve as pointers towards an explanation. The main results of this paper
are contained in Section 3, which explores the sensitivity of weak lensing
results to the modelling of the matter power spectrum on non-linear scales. We
discuss the implications of our findings in Section 4 and summarize our
conclusions in Section 5.

##  2 Preliminaries

###  2.1 The $S_{8}$ tension

In this paper, we focus exclusively on the KiDS-1000 shear-shear results,
adopting their fiducial analysis pipeline and modelling choices777KiDS
Cosmology Analysis Pipeline: https://github.com/KiDS-WL/kcap. Fig. 1 shows the
KiDS21 constraints in the $S_{8}$-$\Omega_{\rm m}$ plane from the publicly
released MCMC chains888http://kids.strw.leidenuniv.nl/sciencedata.php. Asgari
et al. (2021) present results for three two-point statistics: the usual shear
two-point correlation functions $\xi_{\pm}$, COSEBIs (which KiDS21 adopt as
their default for parameter analysis) and angular power spectra. The COSEBIs
and power spectra are estimated by integrating over the correlation functions,
thus any model that fits $\xi_{\pm}$ over the full angular range probed by
KiDS should be consistent with the COSEBIs and power spectra. Fig. 1 also
shows constraints from Planck. The red contours show the constraints from the
DES Y3 shear-shear analysis (Amon et al., 2022a; Secco et al., 2022). As noted
above, $S_{8}$ from DES Y3 is slightly higher but statistically consistent
with the values determined from the KiDS analysis.

The $S_{8}$ tension between Planck and cosmic shear summarized in Sect. 1 is
clearly evident and is particularly acute if one compares Planck TTTEEE+Plens
to the KiDS $\xi_{\pm}$ results. One can also see that the KiDS $\xi_{\pm}$
contours, whilst overlapping with the COSEBI contours, are shifted to lower
values of $\Omega_{\rm m}$. The $\xi_{\pm}$ and the COSEBI analyses differ
primarily in the range of spatial scales sampled by the statistics, with the
$\xi_{\pm}$ contours more sensitive to small scales (Asgari et al., 2021). We
will focus on the KiDS $\xi_{\pm}$ measurements in this paper, though we
discuss the COSEBI measurements in Appendix A.

Figure 1: 68% and 95% constraints in the $S_{8}-\Omega_{\rm m}$ plane for
various data. The blue and navy (dashed) show the constraints from the KiDS
$\xi_{\pm}$ and COSEBI statistics as analyzed by KiDS21, while the red shows
that from the DES Y3 $\Lambda{\rm CDM}$ optimised $\xi_{\pm}$ analysis. The
yellow and grey contours show constraints from Planck TTTEEE with and without
the addition of the Planck CMB lensing likelihood (Plens). The peach contours
labelled EFTofLSS represent constraints from the BOSS power spectrum and
bispectrum effective field theory analysis of D’Amico et al. (2022). The
magenta contours show constraints from redshift space distortions (RSD)
combined with BAO and SN measurements as described in the text. The green
contours show the constraint from the Planck lensing likelihood combined with
BAO together with conservative priors on the acoustic peak location parameter
$\theta_{\rm MC}$ and other cosmological parameters (Planck Collaboration et
al., 2020b).

###  2.2 Constraints from redshift space distortions

Figure 2: Measurements of $f\sigma_{8}$ from various surveys. The green
triangle at $z_{\rm eff}=0.02$ is from an analysis of low redshift Type Ia SN
by Huterer et al. (2017) and the orange square is from the cosmic flows
analysis of Boruah et al. (2020) using SN, surface brightness fluctuations and
Tully-Fisher distances to nearby galaxies. The filled symbols are from the
consensus results from the completed BOSS/eBOSS surveys (Alam et al., 2021).
The measurement at $z=0.15$ is for the SDSS Main Galaxy Sample (Howlett et
al., 2015). The open circles show a reanalysis of the BOSS, eBOSS/LRG and
eBOSS/QSO samples by Brieden et al. (2022) using their ShapeFit methodology.
The open diamond shows results from an effective field theory analysis of the
eBOSS Emission Line Galaxy sample by Ivanov (2021). The grey bands show the
$1\sigma$ and $2\sigma$ regions allowed by the Planck base $\Lambda{\rm CDM}$
cosmology.

Redshift space distortions (RSD) provide an independent way to measure the
growth rate of fluctuations (Kaiser, 1987). RSD provide a measurement of the
parameter combination $f\sigma_{8}$, where $f$ is the logarithmic derivative
of the linear growth rate $D$ with respect to the scale factor

$f={{\rm d\ ln}D\over{\rm d\ ln}a},$ (4)

and $a=(1+z)^{-1}$. In the $\Lambda{\rm CDM}$ model, $f\approx\Omega_{\rm
m}(z)^{0.55}$ (Lahav et al., 1991). RSD measurements from large galaxy
redshift surveys are of particular relevance to the $S_{8}$ tension because
they overlap with the redshift range covered by the KiDS survey. Since most of
the information on RSD from the analyses discussed in this section comes from
wavenumbers in the range $0.01h{\rm Mpc}^{-1}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}k\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.1h{\rm Mpc}^{-1}$ (see, for
example, Fig. 8 of Philcox & Ivanov 2021) corresponding to linear or very
mildly non-linear scales, any discrepancy with the Planck  $\Lambda{\rm CDM}$
model would be fatal to the explanation of the $S_{8}$ tension proposed in
this paper. It is therefore important to review the consistency between RSD
measurements and the Planck $\Lambda{\rm CDM}$ model in detail.

As noted in Planck Collaboration et al. (2016), early RSD measurements used a
fixed fiducial cosmology that differed significantly from the Planck
$\Lambda{\rm CDM}$ cosmology leading to potential biases and underestimates of
the error on $f\sigma_{8}$ (Howlett et al., 2015). In addition, there have
been significant theoretical advances in the modelling of RSD in recent years.
To compute the RSD constraints in Fig. 1 we have used the results shown by the
filled symbols in Fig. 2. The measurements at $z_{\rm eff}=0.38$ and $0.61$
are the ‘consensus’ results from the completed Baryon Oscillation Spectroscopy
Survey (BOSS) and extended BOSS (eBOSS) surveys (Alam et al., 2017,
2021)999SDSS: https://www.sdss.org/science/final-bao-and-rsd/-measurements-
table supplemented by the the measurement from the SDSS Main Galaxy Sample
(MGS) at $z_{\rm eff}=0.15$ (Ross et al., 2015; Howlett et al., 2015) and the
low redshift ($z_{\rm eff}\approx 0.02$) measurements of $f\sigma_{8}$ from
Huterer et al. (2017) and Boruah et al. (2020) from peculiar velocity-density
field correlations (which we loosely bracket under the term ’RSD’ in this
section). The grey contours in Fig. 2 show the $1\sigma$ and $2\sigma$ ranges
contours for the Planck base $\Lambda{\rm CDM}$ cosmology.

The pink contours in Fig. 1 show the constraints in the $S_{8}$-$\Omega_{\rm
m}$ plane derived by combining these RSD measurements (including the BAO
constraints for the BOSS, eBOSS and MGS samples) with the Pantheon supernova
magnitude-redshift relation (Scolnic et al., 2017), Ly$\alpha$-quasar and
Ly$\alpha$-Ly$\alpha$ BAO measurements from (Blomqvist et al., 2019; de Sainte
Agathe et al., 2019) and BAO measurements from the 6dF Galaxy Survey Beutler
et al. (2011). To scale the BAO constraints we impose a Gaussian Planck prior
on the sound horizon, $r_{\rm d}=147.31\pm 0.31\ \text{Mpc}$. The RSD
constraints in Fig. 1 are consistent with the $S_{8}$ results from Planck
TTTEEE. There is also substantial overlap between the RSD and KiDS contours in
Fig. 1.

The consensus RSD measurements plotted in Fig. 2 assume a Planck $\Lambda{\rm
CDM}$ fiducial power spectrum with adjustable BAO location parameters in
addition to the velocity fluctuation amplitude $f\sigma_{8}$. Recently,
several groups have developed ‘full-shape’ analyses based on effective field
theory (EFT) descriptions of non-linear perturbations (e.g. d’Amico et al.,
2020; Ivanov et al., 2020; Chen et al., 2022b; Philcox & Ivanov, 2021; D’Amico
et al., 2022). The EFT analyses aim to constrain cosmological parameters
independently of Planck. However, the nuisance parameters required to model
perturbation theory counter-terms, galaxy biasing, and redshift space
distortions, effectively down-weight information at wavenumbers $k\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}0.2h^{-1}{\rm Mpc}$. As a
consequence of the restricted wavenumber range, the primordial spectral index
$n_{\rm s}$ is poorly constrained in comparison to Planck. Constraining
$n_{\rm s}$ to the Planck best fit value, Ivanov et al. (2020) find
$\sigma_{8}=0.721\pm 0.43$, Philcox & Ivanov (2021) find $\sigma_{8}=0.729\pm
0.040$ and Chen et al. (2022b) find $\sigma_{8}=0.738\pm 0.048$, from fits to
the BOSS power spectra. These are consistent with each other, but sit low
compared to the Planck $\Lambda{\rm CDM}$ estimate, $\sigma_{8}=0.8095\pm
0.0074$, by $1.6-2\sigma$ (see also Tröster et al. (2020)). The recent EFT
analysis of the BOSS power spectrum and bispectrum (D’Amico et al., 2022)
finds $\sigma_{8}=0.794\pm 0.037$ (labelled as $P_{\ell}+B_{0}^{1{\rm
loop}}+B_{2}^{\rm tree}$ in their analysis) in good agreement with both the
Planck $\Lambda{\rm CDM}$ value, as well as the cosmic shear results101010Note
that D’Amico et al. (2022) apply a correction for prior volume effects in the
EFT analysis that bias $\sigma_{8}$ low by about $1\sigma$ if left
uncorrected.. Constraints in the $S_{8}-\Omega_{\rm m}$ plane from this
analysis are shown by the peach contours in Fig. 1.

Recently, Brieden et al. (2021b, a) have developed an extension of the
‘classical’ BOSS/eBOSS analyses, by including an additional parameter that is
sensitive to the parameter combination $\Omega_{\rm m}h$. The constraints on
$f\sigma_{8}$ derived by applying their ShapeFit technique to the BOSS,
eBOSS/LRG and eBOSS/QSO samples (Brieden et al., 2022) are shown by the open
symbols in Fig. 2. These agree almost perfectly with the consensus results
from the BOSS/eBOSS collaborations reported in Alam et al. (2017, 2021).

The open diamond in Fig. 2 shows the constraint $f\sigma_{8}(z_{\rm
eff}=0.85)=0.309^{+0.029}_{-0.041}$ from a full shape EFT analysis of the
eBOSS/ELG sample by Ivanov (2021). This estimate is in strong disagreement
with the Planck base $\Lambda{\rm CDM}$ value (by $\sim 4.7\sigma$) and also
disagrees with the $\Lambda{\rm CDM}$ parameters determined from the BOSS DR12
sample (see their Fig. 1). The Ivanov (2021) eBOSS/ELG estimate of
$f\sigma_{8}$ is clearly an outlier in Fig. 2, suggesting that further work is
required to establish whether the shape of the power spectra is robust to the
large corrections required to account for selection biases in the eBOSS/ELG
sample.

Evidently, there are still small methodological differences in the RSD
analyses that lead to differences of $\sim 1-2\sigma$ in measurements of
$f\sigma_{8}$ inferred from the same data. Although the pink contours in Fig.
1 are consistent with the constraints from Planck , they have substantial
overlap with the KiDS and DES contours. RSD measurements cannot yet
distinguish decisively between the Planck and weak lensing amplitudes
(Efstathiou & Lemos, 2018; Nunes & Vagnozzi, 2021). There is therefore no
compelling evidence from RSD to support claims that linear growth rates are
suppressed at redshifts $z\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}1$
compared to the expectations of $\Lambda{\rm CDM}$ (Macaulay et al., 2013;
Nesseris et al., 2017; Kazantzidis & Perivolaropoulos, 2018; Benisty, 2021;
Abdalla et al., 2022).

Note also that the functional form of the linear growth rate $D(t)$ over the
redshift range $\sim 0.2-0.7$ inferred from a tomographic analysis of cosmic
shear, galaxy clustering and CMB lensing appears to be compatible with that
expected in the $\Lambda{\rm CDM}$ model (García-García et al., 2021), though
the overall fluctuation amplitude is found to be lower than expected in the
Planck base $\Lambda{\rm CDM}$ cosmology.

###  2.3 CMB lensing

Rather than reflecting anomalies in the growth rate of perturbations, the
$S_{8}$ tension may be related to anomalies in the propagation of photons, for
example a departure from General Relativity that results in a ‘gravitational
slip’ (e.g. Daniel et al., 2008; Bertschinger, 2011; Simpson et al., 2013;
Pizzuti et al., 2019). This can be tested using CMB gravitational lensing. The
CMB lensing signal is caused by matter distributed along the line-of-sight
over a broad redshift range with a median redshift of $z\sim 2$. This is not
very much higher than the mean redshift $z\sim 1$ of the highest tomographic
redshift bin in KiDS-1000 (which contributes the highest weight to the KiDS
two point statistics).

As noted in Planck Collaboration et al. (2020b) for base $\Lambda{\rm CDM}$
with ‘lenspriors’ imposed111111These are loose priors of $n_{s}=0.96\pm 0.02$,
$\Omega_{b}h^{2}=0.0222\pm 0.005$ (motivated by the deuterium abundance (Cooke
et al., 2018)), $0.4<h<1$, and the optical depth to reionization fixed at
$\tau=0.055$., the Planck lensing likelihood tightly constrains the parameter
combination121212As this paper was nearing completion Carron et al. (2022)
presented results from an improved Planck  lensing likelihood. This gives
$\sigma_{8}\Omega_{\rm m}^{0.25}=0.599\pm 0.016$, within $0.4\sigma$ of the
Planck  TTTEEE result of Eq. 5b.

$\sigma_{8}\Omega_{\rm m}^{0.25}=0.589\pm 0.020,\quad{\rm Plens},$ (5a) which
is within $1\sigma$ of the constraint derived from the TTTEEE likelihood,
$\sigma_{8}\Omega_{\rm m}^{0.25}=0.6057\pm 0.0081.\quad{\rm TTEEE}.$ (5b)

The addition of BAO data and a conservative prior of $\theta_{\rm
MC}=1.0409\pm 0.0006$131313Where $\theta_{\rm MC}$ is an approximation to the
angular size of the sound horizon at recombination, see Planck Collaboration
et al. (2016) for a definition. (effectively adding another BAO point at
recombination) to the Planck lensing likelihood restricts the range of allowed
values of $\Omega_{\rm m}$ resulting in the green contours141414Computed from
the chains ‘base$\\_$lenspriors$\\_$BAO$\\_$theta’ available from the Planck
Legacy Archive (https://www.cosmos.esa.int/web/planck/pla). in Fig. 1.
Assuming base $\Lambda{\rm CDM}$, the Planck lensing constraints are in very
good agreement with the Planck TTTEEE results (as they should be, since
acoustic peaks of the temperature and polarization power spectra are sensitive
to CMB lensing, Larsen et al. 2016, see e.g.). Over the redshift range probed
by Planck lensing, there is no evidence from Fig. 1 for any gravitational
slip/modified gravity type of effect that might be causing photons to behave
in a different way from the predictions of $\Lambda{\rm CDM}$.

It is possible to test consistency of weak galaxy and CMB lensing at redshifts
$z\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}1$ by cross-correlating
CMB lensing with weak lensing maps and/or maps of the galaxy distribution
(e.g. Marques et al., 2020; Robertson et al., 2021; Krolewski et al., 2021;
Chang et al., 2022; Chen et al., 2022a). We will discuss these measurements
briefly in Sect. 5.

Figure 3: Measurements of KiDS21 two-point correlation functions $\xi_{\pm}$,
scaled by the angular separation, $\theta$ (upper panels). The correlation
functions are measured for each redshift bin pair, indicated by the label and
the error bar represents the square root of the diagonal of the analytic
covariance matrix. The blue line is the best fit $\Lambda{\rm CDM}$
theoretical prediction, including modelling of the non-linear matter power
spectrum using HMCode2020 (very similar to the KiDS21 fit, which used
HMCode2016), which includes a baryon feedback parameter $\Theta_{\rm AGN}$.
The red line fixes cosmological parameters to the Planck best-fit values and
now modulates the non-linear power spectrum using the parameter $A_{\rm mod}$
(Eq. 6). In both cases, the dashed lines show $\xi_{\pm}$ computed from the
linear matter power spectra. The lower panels show the residual differences
between the models and the KiDS21 measurements $\delta\xi_{\pm}/\xi_{\pm}$.
Following KiDS21, scales excluded from the analysis are shaded in green.

##  3 A possible solution

###  3.1 Motivation

Table 1: Mean value of $S_{8}$ and 1$\sigma$ errors from our The entry labelled HMCode2016 uses the identical non-linear matter power spectrum and feedback model as in KiDS21 and reproduces their results to high accuracy. The next set of entries use HMCode2020 to model the non-linear power spectrum applying uniform priors with different ranges for the baryon feedback parameter $\Theta_{\rm AGN}$. (The row labelled ‘no feedback’ has all baryon feedback switched off.) The asterisk denotes the case of a narrow prior on $\Theta_{\rm AGN}$, as assumed by Tröster et al. (2021), which we refer to as the ‘fiducial’ model in this paper. The final set of entries show results for the phenomenological non-linear suppression model governed by the parameter $A_{\rm mod}$ (see Eq.6). For the cases labelled ‘free’, the cosmological parameters are allowed to vary freely. For cases labelled ‘Planck ’ the cosmological parameters are fixed to the TTTEEE values for the Planck best fit $\Lambda{\rm CDM}$ cosmology. The column labelled $N_{\sigma}$ lists $(\chi^{2}_{\rm min}-\sqrt{2N_{\rm deg}})/\sqrt{2N_{\rm deg}}$ where $N_{\rm deg}$ is the effective number of degrees of freedom. We have $N_{\rm deg}=225-4.5$ (accounting for the number of constrained parameters in the fits following Joachimi et al. 2021) for the ‘free’ fits where cosmological parameters are allowed to vary in addition to nuisance parameters (see KiDS21) and $N_{\rm deg}=225-2.5$ for the fits with cosmological parameters constrained to the Planck values. The column labelled PTE gives the probability to exceed the value of $\chi^{2}_{\rm min}$.  Non-linear model & prior range | Cosmology | $S_{8}$ | $\chi^{2}_{\rm min}$ | $N_{\sigma}$ | PTE
---|---|---|---|---|---
HMCode2016 | free | $0.765\pm{0.018}$ | 260.1 | $1.92$ | $0.030$
HMCode2020 no feedback | free | $0.755\pm{0.016}$ | 261.5 | $1.95$ | $0.025$
HMCode2020 $\Theta_{\rm AGN}=7.3-8.3$ | free | $0.774\pm{0.021}$ | 260.2 | $1.89$ | $0.029$
HMCode2020 $\Theta_{\rm AGN}=7.0-10.0$ | free | $0.785\pm{0.030}$ | 260.0 | $1.88$ | $0.030$
HMCode2020 $\Theta_{\rm AGN}=7.0-10.0$ | Planck | $0.829$ | 267.6 | $2.13$ | $0.016$
HMCode2020 $A_{\rm mod}=0.5-1.2$ | free | $0.780\pm{0.035}$ | 260.3 | $1.89$ | $0.029$
HMCode2020 $A_{\rm mod}=0.5-1.2$ | Planck | $0.829$ | 265.5 | $2.04$ | $0.021$

To summarize the previous section:

(i) There is evidence from independent weak lensing surveys that the amplitude
of the matter fluctuations, as measured by the $S_{8}$ parameter, is lower
than expected according to the Planck $\Lambda{\rm CDM}$ cosmology at about
the $2.5-3\sigma$ level.

(ii) Redshift space distortions analyses over the redshift range $0.02-1.6$
are consistent with the Planck base $\Lambda{\rm CDM}$ cosmology, suggesting
that perturbations grow at the rates expected in $\Lambda{\rm CDM}$ over this
entire redshift range.

(iii) CMB lensing is consistent with the expectations of the Planck base
$\Lambda{\rm CDM}$ cosmology, suggesting that photons behave as expected in
this cosmology and that perturbations grow at the expected rates over the
redshift range $z\sim 1000$ to $z\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}2$.

(iv) In addition, the combination of BAO measurements and magnitude-redshift
relation of Type 1a supernovae tightly constrains the expansion history $H(z)$
to be very close to that of the Planck base $\Lambda{\rm CDM}$ cosmology over
the redshift range probed by the weak lensing surveys (Heavens et al., 2014;
Verde et al., 2017; Macaulay et al., 2019; Efstathiou, 2021). Solutions to the
$S_{8}$ tension involving a change in the background cosmology (e.g. invoking
a late time transition to a phantom equation of state as considered by Joudaki
et al. 2017) are disfavoured by the BAO and supernova data.

In this paper we will seek a physical explanation of the $S_{8}$ tension that
ties together points (i)-(iv).

Figure 3 shows the KiDS $\xi_{\rm\pm}$ measurements as reported in KiDS21. The
shaded regions in the $\xi_{-}$ plot shows the range of angular scales that
are excluded from the cosmological analysis by KiDS21 to reduce uncertainties
associated with the modelling of baryonic feedback. We use the public KiDS
pipeline and follow the updated analysis of Tröster et al. (2021), who re-
analyse the KiDS21 band-power measurements with HMCode2020 (Mead et al., 2021)
in place of the earlier version of HMCode (Mead et al., 2015) used in KiDS21.
HMCode2020 provides a description of the non-linear evolution of the matter
power spectrum and models baryonic feedback via a parameter $\Theta_{\rm
AGN}={\rm log}_{10}(T_{\rm AGN}/{\rm K})$ calibrated using the BAHAMAS
(BAryons and HAloes of MAssive Systems) hydrodynamical simulations (McCarthy
et al., 2017; van Daalen et al., 2020), where $T_{\rm AGN}$ is a subgrid
heating parameter. The solid blue line in this figure shows our best fit to
these data applying the same priors on parameters, including $\Theta_{\rm
AGN}$, as in (Tröster et al., 2021, see their Appendix A). The marginalised
mean value of the $S_{8}$ posterior distribution151515Throughout this paper we
always report the marginalised mean value of the posterior distribution and
the 68% confidence limits. is quoted in Table 1 (entry labelled HMCode2020
$\Theta_{\rm AGN}=7.3-8.3$, which we will refer to as the ‘fiducial’
analysis). This is within $0.5\sigma$ of the analysis reported in Table 1
using HMCode2016 (which agrees perfectly with the results of KiDS21),
consistent with the impact on the band powers statistics as reported by
Tröster et al. (2021, Appendix A). The cosmological parameters for this fit
will be discussed further in Sect. 3.3.

The blue dashed lines in Fig. 3 show the predictions for the same parameters
but now using only the linear theory matter power spectrum. One can see that
the non-linear corrections make a dominant contribution to the total signal,
even for $\xi_{+}$ except on angular scales $\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}10-20$ arcmin. Evidently,
extracting precision cosmology from smaller angular scales requires accurate
modelling of the non-linear power spectrum.

Given this sensitivity, we investigate the plausibility of solving the $S_{8}$
tension by modifying the spectrum only in the non-linear regime, while fixing
the linear power spectrum to that of the Planck base $\Lambda{\rm CDM}$
cosmology. With such a solution it may possible to explain the cosmic shear
measurements in a way that is consistent with points (i)-(iv) above, since the
RSD measurements and Planck CMB lensing are insensitive to highly non-linear
scales. In addition, such a solution would not require any modification to the
Planck $\Lambda{\rm CDM}$ expansion history, preserving the consistency of the
$\Lambda{\rm CDM}$ model with the magnitude-redshift relation of Type Ia
supernovae, and the geometrical constraints from BAO measurements.

###  3.2 Phenomenological Model

The lensing data are not yet sufficiently precise to allow a parametric
reconstruction of the matter power spectrum as a function of redshift, so in
this section we adopt a particularly simple phenomenological model. At this
stage, we wish to investigate whether a model with the Planck $\Lambda{\rm
CDM}$ cosmological parameters and linear fluctuation spectrum can provide an
acceptable fit to the KiDS $\xi_{\pm}$ measurements via a modification of the
spectrum in the non-linear regime. We write the matter power spectrum, $P_{\rm
m}(k,z)$, as

$P_{\rm m}(k,z)=P^{\rm L}_{\rm m}(k,z)+A_{\rm mod}[P^{\rm NL}_{\rm
m}(k,z)-P^{\rm L}_{\rm m}(k,z)]\,,$ (6)

where the superscript ${\rm L}$ denotes the linear theory power spectrum, NL
denotes the non-linear power spectrum computed by HMCode2020 with baryonic
feedback switched off and $A_{\rm mod}$ is an amplitude parameter that we vary
in the MCMC analysis.

Fixing the cosmology to the Planck TTTEEE $\Lambda{\rm CDM}$ parameters, but
allowing the ‘nuisance’ parameters describing intrinsic alignments, redshift
calibration errors and $A_{\rm mod}$ to vary, we find the best fit theory
model shown by the solid red line in Fig. 3. As can be seen, this model is
almost indistinguishable from the blue line of the KiDS cosmology. The
posterior distribution of the parameter $A_{\rm mod}$ is shown in red in Fig.
4. We find a mean

$A_{\rm mod}=0.69\pm 0.06\,,$ (7)

and a minimum value of $\chi^{2}_{\rm min}=265.5$ for $225$ data points, which
is very close to the minimum value of $\chi^{2}$ for the fiducial model (see
Table 1). Both models provide acceptable fits to the data. (All of the
$\chi^{2}$ values listed in Table 1 are high by about $2\sigma$ as a
consequence of outliers, for example, the points at $\theta\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}50^{\prime}$ in the (2,2) $\xi_{+}$
correlation function). Note that the uncertainty in Eq. 7 is an underestimate
as we have neglected the error on the Planck base $\Lambda{\rm CDM}$
cosmology. The reanalysis of the KiDS-1000 $\xi_{\pm}$ measurements varying
both the $A_{\rm mod}$ and cosmological parameters is shown as the blue
contour in Fig. 4, highlighting the degeneracy between $S_{8}$ and $A_{\rm
mod}$, which reduces the power of lensing data to constrain $S_{8}$.

We find that fixing the cosmology to Planck does not cause any significant
shifts to the nuisance parameters compared to the fiducial analysis.
Similarly, we have verified that fixing the nuisance parameters to the best
fit values of the fiducial analysis constraints does not alter the constraint
on $A_{\rm mod}$. These tests show that the low value of $A_{\rm mod}$ in Eq.
7 is insensitive to the nuisance parameters.

The results of this section show that a suppression of the non-linear power
spectrum can reconcile the base Planck cosmology with the KiDS lensing data.
However, the model is purely phenomenological and so it is reasonable to ask
whether such a suppression has a physical interpretation. There are two
obvious possibilities: (a) that the suppression is caused by baryonic
feedback; (b) that it is caused by a physical property of the dark matter that
suppresses the power spectrum on small scales, for example some type of self-
interaction (see e.g. the review by Tulin & Yu, 2018) or other non-
gravitational interaction (e.g. Becker et al., 2021), a mixture of cold and
warm dark matter (e.g. Boyarsky et al., 2009), or an axionic dark matter
component with a de Broglie wavelength of a few Mpc (e.g. Hui et al., 2017).
We consider baryonic feedback in the next subsection.

Figure 4: Illustration of the strong degeneracy between $S_{8}$ and the
phenomenological power spectrum suppression parameter $A_{\rm mod}$. The blue
contours shown in the left-hand panel represent the 68% and 95% constraints
for the KIDS $\xi_{\pm}$ statistics analyzed by using HMCode2020-no feedback,
but instead varying $A_{\rm mod}$. The dashed line indicates the Planck
$\Lambda{\rm CDM}$ best-fit value for the $S_{8}$ parameter. The posterior of
$A_{\rm mod}$ for this case is shown by the blue curve in the right hand
panel. If the cosmological parameters are fixed to the Planck $\Lambda{\rm
CDM}$ best-fit values we find the posterior for $A_{\rm mod}$ shown by the red
curve in the right hand panel.

###  3.3 Baryonic Feedback

Figure 5: 68% and 95% constraints for KIDS $\xi_{\pm}$ statistics analyzed
using HMCode2020, with varying prior ranges for $\Theta_{\rm AGN}$, compared
to the HM2016 analysis by KiDS21 (yellow). Following Tröster et al. (2021), we
allow $\Theta_{\rm AGN}$ to vary with a uniform prior over the range $7.3-8.3$
(blue). The range is extended to $7.1-10.0$ (green) to allow for more extreme
feedback. The constraints neglecting feedback are shown in black dotted. We
fix the cosmological parameters to Planck $\Lambda{\rm CDM}$ best fit values
and find the posterior on $\Theta_{\rm AGN}$ shown by the dashed red line in
the bottom right panel. The dashed lines in the other panels show the Planck
$\Lambda{\rm CDM}$ best-fit values for $\Omega_{\rm m}$, $\sigma_{8}$ and
$S_{8}$.

Hydrodynamic simulations of structure formation have improved remarkably over
the last decade (e.g. Schaye et al., 2015; McCarthy et al., 2017; Nelson et
al., 2019). Nevertheless, the detailed physics involved in baryon cooling,
star formation, stellar feedback and feedback from active galactic nuclei
(AGN) remains extremely uncertain. As pointed out by many authors, feedback
process redistribute the baryons leading to a suppression of the dark matter
power spectrum in the non-linear regime (see e.g. van Daalen et al., 2011;
Vogelsberger et al., 2014b; Hellwing et al., 2016; Chisari et al., 2019, and
references therein). Since feedback processes are not well understood, there
is considerable uncertainty in the overall amplitude of this suppression, its
scale dependence and evolution with redshift. In this section we use the
feedback scheme implemented in HMCode2020 and ask how far do we have to vary
the feedback parameter $\Theta_{\rm AGN}$ to match the suppression of our
phenomenological model of Eq. 7.

We first pin the cosmological parameters to the Planck TTTEEE $\Lambda{\rm
CDM}$ parameters then fit the KiDS $\xi_{\pm}$ data using HMCode2020 with a
broad prior161616This test is similar to that of Yoon & Jee (2021), who use
HMCode2016 and KiDS-450. on $\Theta_{\rm AGN}$ that is uniform in the range
$7.0<\Theta_{\rm AGN}<10.0$ . This prior range is substantially wider than the
range $7.6<\Theta_{\rm AGN}<8.0$ over which HMCode2020 was calibrated against
the BAHAMAS simulations. As we move outside the calibrated regime, the link
between the parameter $\Theta_{\rm AGN}$ and physical models of feedback
becomes more tenuous.

The posterior distribution of the $\Theta_{\rm AGN}$ parameter is shown in red
dashed in Fig. 5 and has a marginalised mean of

$\Theta_{\rm AGN}=9.17^{+0.2}_{-0.8}\,.$ (8)

As in the previous section, we have verified that fixing the cosmology to
Planck $\Lambda{\rm CDM}$ does not lead to significant differences in the
nuisance parameters compared to the fiducial analysis and that fixing the
nuisance parameters to their fiducial best-fit values does not change the
constraint of Eq. 8. This model gives a minimum value of $\chi^{2}_{\rm
min}=267.6$ (see Table 1), almost identical to the minimum value of $\chi^{2}$
for the $A_{\rm mod}$ model with cosmological parameters fixed to Planck. Note
that the error in Eq. 8 does not include the uncertainty in the Planck base
$\Lambda{\rm CDM}$ cosmology. As expected, HMCode2020 with a broad prior on
$\Theta_{\rm AGN}$ behaves qualitatively like the phenomenological $A_{\rm
mod}$ model.

Next, we fit the $\xi_{\pm}$ measurements allowing the cosmological parameters
to vary while extending the prior range for $\Theta_{\rm AGN}$ in HMCode2020.
We have also performed a fit with baryonic feedback switched off. The
parameter constraints for these fits are shown in Fig. 5. Table 1 and gives
values for $S_{8}$ and $\chi^{2}$ for these fits. Evidently, the parameters
are degenerate and opening up the prior on $\Theta_{\rm AGN}$ leads to long
tails extending to high values of $S_{8}$. Fig. 5 illustrates clearly how high
values of $S_{8}$ are disfavoured by the assumed prior on the baryonic
feedback model adopted by Tröster et al. (2022).

##  4 Comparison with the effects of baryonic feedback in numerical
simulations

Figure 6:  The suppression of the matter power spectrum, $P_{\rm m}(k)/P_{\rm
DMonly}(k)$, required to match the KiDS $\xi_{\pm}$ measurements assuming the
Planck $\Lambda{\rm CDM}$ cosmology. The upper and lower panels shows the
suppression at $z=0$ and $z=1$ respectively. The left hand panels show the
HMCode2020 model of Eq. 8 and the right hand panel shows the phenomenological
model of Eq. 7. The grey bands show the $1\sigma$ allowed ranges. The coloured
lines, taken from van Daalen et al. (2020), show the suppression measured from
cosmological hydrodynamical simulations incorporating baryonic feedback. The
sources are as follows: Illustris (yellow; Vogelsberger et al., 2014a),
Illustris TNG300 (green; Springel et al., 2018), Horizon (purple; Dubois et
al., 2014), C-OWLS (red; COSMO-Overwhelmingly Large Simulations Le Brun et
al., 2014) and BAHAMAS (blue; McCarthy et al., 2017) for several values of
their AGN subgrid heating parameters, ${\rm log}_{10}(\Delta T_{\rm heat}/{\rm
K})$.

Very little is known about the dark matter in the Universe. Therefore, it is
possible that a resolution of the $S_{8}$ tension along the lines of our
$A_{\rm mod}$ parameter is reflective of new physics in the dark sector.
However, as the landscape of such new physics is enormous and highly
speculative, it is reasonable to investigate first whether the non-linear
power suppression required to resolve the $S_{8}$ tension can be realized
using less exotic physics. The posteriors of the $\Theta_{\rm AGN}$ parameter
in Fig. 5 show that the KiDS weak lensing data alone are not able to constrain
both cosmological parameters and the HMCode2020 baryonic feedback model.
Additional information is therefore necessary to quantify the effects of
baryonic feedback on cosmic shear. In this Section we discuss whether the non-
linear power spectrum suppression required to reconcile the Planck
$\Lambda{\rm CDM}$ cosmology with KiDS weak lensing is compatible with
hydrodynamic numerical simulations that include feedback processes.

The power spectrum suppression required by Eq. 8 is shown by the heavy black
lines on the the left-hand side of Fig. 6, together with $1\sigma$ error
bands. The suppression required by the phenomenological model, Eq. 7, is shown
in the right-hand panels. The various coloured lines show the suppression of
the matter power spectrum caused by baryonic feedback as predicted by a number
of cosmological hydrodynamical simulations at $z=0$ and $z=1$. The simulation
results are from the library171717https://powerlib.strw.leidenuniv.nl compiled
by van Daalen et al. (2020), comprised of the original sources cited in the
caption of Fig. 6. For both models, the power spectrum suppression required to
reconcile the KiDS cosmic shear measurements with the Planck cosmology is more
extreme than predicted by most of the hydrodynamic simulations.181818The DES22
$\Lambda{\rm CDM}$-Optimised analysis shown in Fig. 1 gives a slightly higher
value of $S_{8}$ compared to the KiDS analysis but is lower than the Planck
$\Lambda{\rm CDM}$ value (as in all previous cosmic shear analyses). DES22
makes different analysis choices compared to KiDS21 e.g. to reduce sensitivity
to baryonic feedback (Krause et al., 2021). The DES approach is to apply more
conservative angular scale cuts designed to mitigate feedback as predicted by
C-OWLS AGN (red dot-dashed line in Fig. 6). A detailed analysis is therefore
required to compare and combine the KiDS21 and DES22 results. Nevertheless it
is likely that a less extreme non-linear correction is required to explain the
DES $S_{8}$ tension compared to KiDS. At $z=1$, only the C-OWLS simulations
with high values of the AGN feedback parameter can match the suppression
required in our models, though these models fail to reproduce the local gas
fractions in groups and clusters (McCarthy et al., 2017).

Is the power spectrum suppression required to alleviate the $S_{8}$ tension
too extreme for baryonic feedback? Chisari et al. (2019) provide a
comprehensive review of the modelling of baryonic feedback mechanisms in
cosmological simulations and their effects on the matter power spectrum. The
fiducial BAHAMAS model adopts $\Theta_{\rm AGN}=7.8$, since this value
reproduces the gas mass fractions in groups and clusters and the galaxy
stellar mass function at $z\sim 0.1$ (McCarthy et al., 2017). The prior of
$7.3<\Theta_{\rm AGN}<8.3$ used by Tröster et al. (2022) is broader than the
range over which HMCode2020 has been calibrated against the BAHAMAS
simulations and is intended to place conservative bounds on physically
reasonable models of feedback.

However, as noted by Chisari et al. (2019) baryonic feedback processes are
complex and poorly understood. It is perhaps possible that hydrodynamical
simulations do not capture the complexities of feedback. For example, it is
not feasible to model processes such as black hole growth, magnetic fields,
jet formation, cosmic ray injection ab initio. (see e.g. Enßlin et al., 2011;
Hopkins et al., 2021; Beckmann et al., 2022). It is also not clear whether
simulations calibrated on the local gas fractions of groups and clusters can
be extrapolated to redshifts $z\sim 0.5-1$ relevant to the interpretation of
cosmic shear surveys. It is therefore important to develop empirical ways of
constraining baryon feedback rather than relying on numerical simulations.

Joint analyses and cross-correlations between cosmic shear and the kinetic and
thermal Sunyaev-Zeldovich (SZ) effects (e.g. Schneider et al., 2021; Tröster
et al., 2022) offer a promising way of testing baryonic feedback. In
particular, SZ observations are sensitive to baryons on larger spatial scales
than X-ray observations. Schneider et al. (2021) use the shear power spectrum
measurements from KiDS21, Atacama Cosmology Telescope (ACT) measurements of
the kinetic SZ (Schaan et al., 2021), together with gas and stellar fractions
in groups and clusters inferred from X-ray measurements (Giri & Schneider,
2021) to reconstruct the non-linear suppression of power using their
baryonification model of feedback. If they constrain the baryon fraction to be
within the range allowed by the CMB, they find a power suppression (their Fig.
7) that is extreme than most of the simulations, including BAHAMAS. In
particular, they find evidence for a significant suppression of power at the
wavenumbers $\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}1\,h{\rm
Mpc}^{-1}$ in good agreement with our best fit $A_{\rm mod}$ model. The cross-
correlation analysis of Tröster et al. (2022) of the Planck thermal SZ and
shear power spectrum measurements supports high values of $\Theta_{\rm AGN}$
that are more extreme than predictions from BAHAMAS, but are restricted by the
prior range adopted for $\Theta_{\rm AGN}$ (a similar effect can be seen in
Fig. 5). Hints for more extreme feedback have also been found from other
analyses that utilise ACT SZ information Amodeo et al. (2021); Pandey et al.
(2021).

To summarize, our analysis shows that a suppression of power extending to
large scales corresponding to wavenumbers of $k\sim 0.2\ h{\rm Mpc}^{-1}$ is
required if the Planck $\Lambda{\rm CDM}$ cosmology is to be reconciled with
cosmic shear. Such strong feedback on large scales is not seen in most of the
hydrodynamic simulations shown in Fig. 6. If such extreme baryonic feedback
can be ruled out, then a non-linear solution to the $S_{8}$ tension would
probably require new dark matter physics.

##  5 Conclusions

This paper is motivated by the substantial evidence from galaxy lensing
measurements that the parameter $S_{8}$ has a lower value than expected in the
Planck $\Lambda{\rm CDM}$ cosmology. Noting that the signal-to-noise driving
the cosmic shear constraints is dominated by non-linear scales, we have
investigated whether the Planck $\Lambda{\rm CDM}$ cosmology can be reconciled
with these measurements by modifying the matter power spectrum on non-linear
scales, preserving all other features of $\Lambda{\rm CDM}$. If this
explanation is correct:

$\bullet$ growth rate measurements that are sensitive mainly to linear scales,
in particular, precision measurements of RSD from wavenumbers $k\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.1\,h{\rm Mpc}^{-1}$ using
forthcoming data from the Dark Energy Spectroscopic Instrument (DESI; DESI
Collaboration et al., 2016), should agree with the Planck cosmology.

$\bullet$ the background expansion history, $H(z)$, and consequently
luminosity and angular diameter distances, should agree with the Planck
$\Lambda{\rm CDM}$ cosmology at all redshifts;

$\bullet$ photons should respond to gravity as expected in General Relativity.
Gravitational lensing measurements (based on either the CMB or galaxies) that
are sensitive primarily to linear scales, should agree with the Planck
$\Lambda{\rm CDM}$ cosmology at all redshifts;

A detailed analysis of the two-point correlation functions $\xi_{\pm}$, as
measured by KiDS21, shows that the Planck $\Lambda{\rm CDM}$ model can provide
acceptable fits if the power-spectrum on non-linear scales is suppressed via
our phenomenological model of Eq. 4, with a value $A_{\rm mod}\approx 0.69$.
If this suppression is interpreted as being caused by baryonic feedback, then
a comparison with numerical hydrodynamic simulations shows that strong
baryonic feedback is required. The amplitude and spatial extent of the
suppression that we require are well outside the ranges found in the BAHAMAS
simulations and cannot be reproduced by baryonic feedback models adopted in
KiDS21 and Tröster et al. (2021) given their choices of priors. However, the
physics of baryonic feedback is extremely complex and multi-scale and it may
be premature to exclude models with strong baryonic feedback. As mentioned in
Sect. 3.3, combinations of cosmic shear with thermal and kinetic SZ
measurements provide hints that baryonic feedback may be stronger than
conventionally thought (Amodeo et al., 2021; Schneider et al., 2021; Tröster
et al., 2022). In particular, kinetic SZ measurements offer the possibility of
constraining the effects of baryonic feedback into the mildly non-linear
regime.

Recently some evidence for low values of $S_{8}$, though not at high
statistical significance, has come from cross-correlations of CMB lensing with
photometric galaxy catalogues (Marques et al., 2020; Robertson et al., 2021;
Hang et al., 2021; Krolewski et al., 2021; Chang et al., 2022; White et al.,
2022), or with galaxy redshift surveys (Chen et al., 2022a). In some of these
analyses, modifications to the non-linear power spectrum as proposed in this
paper may increase the inferred values of $S_{8}$. For others (e.g. Chen et
al., 2022a) the low values of $S_{8}$ are driven by cross-correlations at
large angular scales that are insensitive to the non-linear power spectrum but
may be affected by selection biases. An analysis of galaxy-lensing cross-
correlations shows that the preference for low values of $S_{8}$ comes from
measurements at small scales (Amon et al., 2022a).

It is important to recognise the possibility that a suppression of the matter
power spectrum on non-linear scales may be a consequence of new physics in the
dark sector. For example, a suppression would arise if a fraction of the dark
matter were in the form of a light axionic particle with a de Broglie
wavelength of a few Mpc (Widrow & Kaiser, 1993; Hu et al., 2000; Hui et al.,
2017) though there are many other possibilities as discussed by Hooper et al.
(2022). More detailed investigations of such models would be worthwhile,
particularly if it can be demonstrated definitively that baryonic feedback
cannot solve the $S_{8}$ tension. In the longer term, it may be possible to
differentiate between baryonic feedback and the physics of dark matter by
measuring the redshift dependence of the matter power spectrum suppression
(Viel et al., 2013) and using the high redshift reach of the quasar
Lyman-$\alpha$ forest as discussed by Hooper et al. (2022).

## Acknowledgements

We are particularly grateful to Catherine Heymans and Hendrik Hildebrandt for
a careful reading of a draft of this paper. We thank Florian Buetler, Oliver
Philcox, Licia Verde and Zvonomir Vlah for correspondence on redshift space
distortions and Guido D’Amico, Leonardo Senatore and collaborators, for
sharing their MCMC chains that we used in Fig. 1. We indebted to Joe Zuntz for
developing and maintaining the cosmosSIS framework (Zuntz et al., 2015).

AA receives support from a Kavli Fellowship at Cambridge University. 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).

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##  Appendix A COSEBIs

Figure 7: Measurements of KiDS21 COSEBI coefficients, $E_{\rm n}$, together
with $1\sigma$ error bars. The numbers in each panel denote the photometric
redshift bins used in each cross-correlation as in Fig. 3 and a zero line is
shown for reference (black dotted). The blue line is the best fit $\Lambda{\rm
CDM}$ theoretical prediction, including modelling of the non-linear matter
power spectrum using HMCode2020 (very similar to the KiDS21 fit, which used
HMCode2016), which includes a baryon feedback parameter $\Theta_{\rm AGN}$ as
in our fiducial analysis. The dashed lines indicate the predictions based upon
only the linear matter power spectra.

The fiducial analysis in KiDS21 is based on the COSEBI statistic $E_{n}$. The
COSEBI statistics are constructed from linear combinations of $\xi_{\pm}$ to
separate the signals from $E$\- and $B$-mode lensing:

$\displaystyle E_{n}$ $\displaystyle=$ $\displaystyle{1\over
2}\int_{\theta_{\rm min}}^{\theta_{\rm
max}}[T_{+n}(\theta)\xi_{+}(\theta)+T_{-n}(\theta)\xi_{-}(\theta)]\,,$ (9a)
$\displaystyle B_{n}$ $\displaystyle=$ $\displaystyle{1\over
2}\int_{\theta_{\rm min}}^{\theta_{\rm
max}}[T_{+n}(\theta)\xi_{+}(\theta)-T_{-n}(\theta)\xi_{-}(\theta)]\,,$ (9b)

where $n$ is an integer and the filter functions $T_{\rm n}$ depend on the
choices of $\theta_{\rm min}$ and $\theta_{\rm max}$. Weak lensing should
generate pure $E$ modes and so the detection of a B-mode signal is a signature
of systematics in the cosmic shear catalogues.

For KiDS-1000, the $B$-mode signal is consistent with zero. KiDS21 chose to
analyze the first five $n$-modes of $E_{n}$, as plotted in Fig. 7. The
weighting of wavenumbers in the statistic $E_{n}$ depends on the range of
$\theta_{\rm min}$ and $\theta_{\rm max}$. For the choices in KiDS21,
$0.5^{\prime}\leq\theta\leq 300^{\prime}$, the $E_{n}$ statistic is gives
lower weight to high wavenumbers in comparison to their $\xi_{\pm}$ data
vector and is therefore less sensitive to their baryonic feedback model and
other small-scale systematics (Asgari et al., 2020).

However, the $E_{n}$ statistic does retain sensitivity to non-linear scales
and to baryonic physics, as is illustrated in Fig. 8 and Fig. 7. The solid
line in Fig. 7 shows the best fit to $E_{n}$ using HMCode2020) with our
fiducial prior on $\Theta_{\rm AGN}$. The dashed line shows the prediction
based upon the linear power spectrum for this fit. One can see that the
sensitivity to non-linearities (and therefore to baryonic feedback), for their
choices of $\theta_{\rm min}$ and $\theta_{\rm max}$, is restricted almost
exclusively to the $E_{1}$ coefficients which carry much higher statistical
weight in the likelihood than the coefficients with $n>1$. As a consequence,
the $E_{n}$ statistic has very little shape discrimination. Figure 8
illustrates that the COSEBIS and $\xi_{\pm}$ show a similarly strong
degeneracy between $A_{\rm mod}$ and $S_{8}$, though the COSEBI contours are
considerably wider than those for $\xi_{\pm}$.

Another consequence of using COSEBIs is that they introduce a correlation
between $S_{8}$ and $\Omega_{\rm m}$ that extends to low values of $S_{8}$ at
high values of $\Omega_{\rm m}$ that are strongly disfavoured by Planck. As
discussed by KiDS21, the COSEBIs constraints on the parameter combination
$\Sigma_{8}=\sigma_{8}(\Omega_{\rm m}/0.3)^{0.54}$ are almost independent of
$\Omega_{\rm m}$. In assessing consistency with Planck $\Lambda{\rm CDM}$ it
is better to use the posterior distribution of $\Sigma_{8}$ rather than
$S_{8}$. Similar remarks apply to the Tröster et al. (2022) analysis, which is
based on the lensing power spectrum band powers.

Figure 8: Version of Fig. 4 illustrating the strong degeneracy between $S_{8}$
and the phenomenological power spectrum suppression parameter $A_{\rm mod}$
but including results from the COSEBIS statistics. The left hand panel shows
the constraints from on $A_{\rm mod}$ and $S_{8}$ allowing cosmological and
nuisance parameters to vary but with baryonic feedback set to zero. The solid
blue contours show the constraints using the COSEBIS, while the dotted blue
contours show the results based on $\xi_{\pm}$ (which are the same as those
plotted in Fig. 4). The dashed line indicates the Planck $\Lambda{\rm CDM}$
best-fit value for the $S_{8}$ parameter. The posteriors of $A_{\rm mod}$ are
shown by the blue and dashed curves in the right hand panel. If the
cosmological parameters are fixed to the Planck $\Lambda{\rm CDM}$ best-fit
values we find the posterior for $A_{\rm mod}$ shown by the red curves in the
right hand panel.