# First measurement of the Weyl potential evolution from the Year 3 Dark
Energy Survey data: Localising the $\sigma_{8}$ tension

Isaac Tutusaus<EMAIL_ADDRESS>Institut de Recherche en
Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES,
14 Av. Edouard Belin, 31400 Toulouse, France  Camille Bonvin
<EMAIL_ADDRESS>Departement de Physique Theorique and Center for
Astroparticle Physics, Universite de Geneve, Quai E. Ansermet 24, CH-1211
Geneve 4, Switzerland  Nastassia Grimm<EMAIL_ADDRESS>Departement
de Physique Theorique and Center for Astroparticle Physics, Universite de
Geneve, Quai E. Ansermet 24, CH-1211 Geneve 4, Switzerland

###### Abstract

We present the first measurement of the Weyl potential at four redshifts bins
using data from the first three years of observations of the Dark Energy
Survey (DES). The Weyl potential, which is the sum of the spatial and temporal
distortions of the Universe’s geometry, provides a direct way of testing the
theory of gravity and the validity of the $\Lambda$CDM model. We find that the
measured Weyl potential is 2.3$\sigma$, respectively 3.1$\sigma$, below the
$\Lambda$CDM predictions in the two lowest redshift bins. We show that these
low values of the Weyl potential are at the origin of the $\sigma_{8}$ tension
between Cosmic Microwave Background (CMB) measurements and weak lensing
measurements. Interestingly, we find that the tension remains if no
information from the CMB is used. DES data on their own prefer a high value of
the primordial fluctuations, followed by a slow evolution of the Weyl
potential. A remarkable feature of our method is that the measurements of the
Weyl potential are model-independent and can therefore be confronted with any
theory of gravity, allowing efficient tests of models beyond General
Relativity.

##  I Introduction

Since the observation of the accelerated expansion of the Universe in 1998
SupernovaSearchTeam:1998fmf ; SupernovaCosmologyProject:1998vns , extensive
studies have been performed to understand if this acceleration is due to a
cosmological constant, a dynamical dark energy, or a modification of the
theory of gravity. In theories beyond $\Lambda$CDM, not only the background
evolution of the Universe can be modified, but also the formation of
structure. A powerful way of testing these theories is therefore to confront
them with measurements from large-scale structure surveys.

The landscape of theories beyond $\Lambda$CDM is vast, and testing all models
one by one has become infeasible. As a consequence, two approaches have been
developed over the years to efficiently probe deviations from a cosmological
constant. The first one consists in testing classes of theories, e.g.
Horndeski theories Horndeski1974  that encompass all scalar-tensor theories
with second-order equations of motion. The advantage of such an approach is
that the functions constrained by the data are directly linked to fundamental
ingredients in the theories. However, the degeneracies between the free
functions are large, and current data are not stringent enough to constrain
them all well, see e.g. Noller:2018wyv .

The second approach is more phenomenological and consists in parameterising
deviations from General Relativity (GR) directly at the level of Einstein’s
equations by introducing two new functions, often called $\mu$ and $\eta$, see
e.g. Koyama:2015vza . The function $\mu$ encodes changes in Poisson’s
equation, while $\eta$ describes the difference between the time distortion
and the spatial distortion of the metric, the so-called anisotropic stress.
This approach has been successfully used with current data, e.g. combining
gravitational lensing measurements from the Dark Energy Survey (DES) and
galaxy clustering measurements from BOSS and eBOSS DES:2022ygi . Even though
powerful, this approach suffers from one important complication: the
constraints on $\mu$ and $\eta$ at a given redshift $z$ depend on the
evolution of the functions at _all_ redshifts above $z$. This means that
either one needs to assume a given time evolution for these functions, as done
in DES:2022ygi , in which case only the values of $\mu$ and $\eta$ at $z=0$
are constrained and the time evolution is fixed. Or one needs sophisticated
techniques to constrain and reconstruct the time evolution of $\mu$ and $\eta$
from a set of chosen redshift nodes, as done for example in Bonvin:2022tii .
In this second case, it is necessary to have measurements over a wide range of
redshift to obtain relevant constraints, and any degradation of the data in a
given redshift bin will impact the constraints in the other bins. Another
limitation of this framework is that the constraints on $\mu$ and $\Sigma$
rely on the validity of Euler’s equation for dark matter. Without this
additional assumption, $\mu$ cannot be constrained from redshift-space
distortions (RSD), since it is fully degenerated with the parameters encoding
changes in Euler’s equation SvevaNastassiaCamille ; Bonvin:2022tii . As a
consequence, $\Sigma$ can also not be constrained if Euler’s equation is not
valid.

In this paper, we propose an alternative approach, which is fully detached
from any theoretical model. We ask ourselves: what are the quantities that we
can directly measure from data, in a fully model-independent way? This
approach is actually not new and it has been extensively used in the past for
RSD. Combining the multipoles of the RSD correlation function (or power
spectrum) provides direct measurements of the galaxy peculiar velocities in
the redshift bins of the surveys. The evolution of the velocity is encoded in
the so-called growth rate function, $f$, which is measured in combination with
$\sigma_{8}$ (the amplitude of density perturbations in spheres of 8 Mpc/$h$)
Song:2008qt ; Blake_2011 ; eBOSS:2020yzd . Such measurements are very
powerful, since $f\sigma_{8}$ can then be compared with the prediction from GR
to see if this theory is consistent or not. Moreover, it can be compared with
predictions from any other theory of gravity, to put constraints on their
parameters, see e.g. eBOSS:2020yzd .

To the best of our knowledge, this approach, which is standard in RSD
analyses, has never been used for gravitational lensing analyses with real
observations before. The goal of this paper is to do precisely that, using DES
measurements of galaxy-galaxy lensing and galaxy clustering. More precisely,
we use the formalism developed in Tutusaus:2022cab  to measure the evolution
of the Weyl potential $\Psi_{W}=(\Phi+\Psi)/2$ in four redshift bins, which
correspond to the tomographic bins for the DES lenses. This provides the first
direct and model-independent measurement of the evolution of the perturbed
geometry of our Universe. We define a new function, $\hat{J}$, that we call
the _Weyl evolution_ , to encode the evolution of the Weyl potential. The
measurement of $\hat{J}$ from DES data is fully complementary to the growth
rate measurement. Modified theories of gravity can indeed change the way
structures evolve, the way the geometry evolves, or both. As for the
measurement of $f\sigma_{8}$, the Weyl evolution $\hat{J}$ is measured
redshift bin per redshift bin, without assuming any time evolution.

We find that $\hat{J}$ can be measured with a precision of $4-9\%$ when
$3\sigma$ priors from Planck are used to constrain early time cosmological
parameters. These constraints mildly degrade to $5-11\%$ when only DES data
are used. Imposing a more stringent scale cut on the angular correlation
functions further degrades the constraints by a factor of 2. In the baseline
analysis (i.e. with Planck priors and standard scale cuts), we find that the
measurements of the Weyl evolution are 2.7$\sigma$, respectively 3.1$\sigma$,
below the predictions from $\Lambda$CDM in the first and second redshift bins
of DES. Assuming that $\Lambda$CDM is valid, these measurements can be used to
infer the value of $\sigma_{8}$ today. Using priors from Planck we recover the
well-known tension between CMB and weak lensing, which we find to be
2.5$\sigma$. More surprisingly, we find that the tension between the high
redshift clustering amplitude and the low redshift clustering amplitude
remains when no information from Planck is used.

Finally, we compare our measurements of $\hat{J}$ with various models of
modified gravity. The flexibility of our method allows us to compare different
time evolutions of the modified gravity parameters. While the constraining
power of current data is too small to distinguish between various time
evolutions, the coming generation of surveys like Euclid and LSST is expected
to provide more stringent measurements of $\hat{J}$ over a wider redshift
range, potentially allowing to reconstruct the time evolution using our
method.

##  II The lensing angular power spectra as function of the Weyl evolution

The formalism to relate gravitational lensing observables to the Weyl
evolution $\hat{J}$ has been derived in Tutusaus:2022cab . Here we summarise
the main results.

We work with the linearly perturbed flat Friedmann-Lemaître-Robertson-Walker
(FLRW) metric in the conformal Newtonian gauge, with the line element given by

$\displaystyle\mathrm{d}s^{2}=a^{2}(\tau)\Big{[}-(1+2\Psi)\mathrm{d}\tau+(1+2\Phi)\mathrm{d}\mathbf{x}^{2}\Big{]}\,,$
(1)

where $\tau$ denotes conformal time, $a$ is the scale factor, and $\Psi$ and
$\Phi$ are the two metric potentials. Gravitational lensing is directly
sensitive to the Weyl potential $\Psi_{W}\equiv(\Phi+\Psi)/2$. We define the
Weyl transfer function $T_{\Psi_{W}}$, which relates the value of
$\Psi_{W}(z,\mathbf{k})$ in Fourier space and at redshift $z$ to the
primordial potential at the end of inflation $\Psi_{\rm in}(\mathbf{k})$. In
GR, one can easily show that for wavelengths that are inside the horizon, the
Weyl transfer function simply evolves proportionally to $D_{1}(z)\Omega_{\rm
m}(z)$. Here $D_{1}$ is the growth function (governing the evolution of matter
density perturbations) and $\Omega_{\rm m}(z)$ is the matter density parameter
at redshift $z$. Hence, in GR, measuring the evolution of density
perturbations or measuring the evolution of the Weyl potential provides
exactly the same information. In modified theories of gravity the situation is
different: Einstein’s equations are modified, generically leading to a
different evolution of the Weyl potential. To capture this, we introduce a new
function, $J$, encoding this evolution. More precisely, we write the transfer
function of the Weyl potential as (see Tutusaus:2022cab  for more detail
111Note that there is typo in Tutusaus:2022cab : $\sqrt{B(k,z_{*})}$ is
missing in the denominator. This does not change any of the results therein.)

$\displaystyle T_{\Psi_{W}}(k,z)=$
$\displaystyle\frac{\mathcal{H}^{2}(z)J(k,z)}{\mathcal{H}^{2}(z_{*})D_{1}(z_{*})}\frac{\sqrt{B(k,z)}}{\sqrt{B(k,z_{*})}}T_{\Psi_{W}}(k,z_{*})\,,$
(2)

where $z_{*}$ is a redshift well in the matter era (before the acceleration of
the Universe started), $\mathcal{H}$ is the Hubble parameter in conformal
time, and $B(k,z)$ is a boost factor, encoding the non-linear evolution of
matter density perturbations at small scales. We assume that at $z_{*}$ a cold
dark matter (CDM) Universe, governed by GR, is recovered. This assumption
automatically excludes early dark energy models, where deviations from a CDM
Universe take place already at high redshift. The motivation behind this
assumption is that, at early times, CMB measurements are highly compatible
with a CDM Universe Planck:2018vyg . Theories of modified gravity therefore
usually aim at recovering GR at high redshift, while leading to a phase of
accelerated expansion at low redshift. In our framework, any deviation in the
Weyl potential evolution during the phase of accelerated expansion is encoded
in the free function $J$. By measuring this free function as a function of
redshift, we will be able to reconstruct the Weyl evolution and assess its
compatibility with GR predictions, or with the predictions of any theory of
gravity we may want to test. Note that the function $J$ has the same
information content as the function $L$ defined in Amendola:2012ky ;
Amendola:2013qna , which identified for the first time the quantities that can
be measured in a model-independent way from large-scale structure surveys.

In full generality, $J$ is a function of redshift and wavenumber $k$. In GR,
the evolution is almost scale-independent, except from a small scale-
dependence due to massive neutrinos. In our analysis we fix the sum of
neutrino masses to 0.06 eV, leading to a scale-dependence in the growth of
structure (and therefore in $J$) which is negligible Euclid:2019clj . In
models of gravity beyond GR, $J$ can in principle depend on scales. However,
in many models of modified gravity, it turns out that the evolution is scale-
independent for sub-horizon modes, where the quasi-static approximation is
valid Gleyzes:2015rua ; Raveri:2021dbu . In our analysis, we therefore drop
the scale-dependence in $J$ (as is also usually done for measurements of the
growth rate in RSD analyses Song:2008qt ; Blake_2011 ; eBOSS:2020yzd ). This
simplifies the analysis, but it is not a fundamental limitation of the method,
which can be extended to a scale-dependent $J$.

As observable we consider the galaxy-galaxy lensing angular power spectrum,
i.e. the cross-correlation of cosmic shear of background galaxies with
clustering of foreground lenses. As shown in Tutusaus:2022cab , the galaxy-
galaxy lensing angular power spectrum can be written in the following way
(using the Limber approximation):

$\displaystyle C_{\ell}^{\Delta\kappa}(z_{i},z_{j})=$
$\displaystyle\frac{3}{2}\int\text{d}z\,n_{i}(z)\mathcal{H}^{2}(z)\hat{b}_{i}(z)\hat{J}(z)B\left(k_{\ell},\chi\right)\frac{P_{\delta\delta}^{\rm
lin}\left(k_{\ell},z_{*}\right)}{\sigma_{8}^{2}(z_{*})}\int\text{d}z^{\prime}n_{j}(z^{\prime})\frac{\chi^{\prime}(z^{\prime})-\chi(z)}{\chi(z)\chi^{\prime}(z^{\prime})}\,,$
(3)

where $n_{i}$ and $n_{j}$ denote the galaxy distribution function of the
lenses and sources, respectively, and $k_{\ell}\equiv(\ell+1/2)/\chi$, with
$\chi$ the comoving distance. $P_{\delta\delta}^{\rm lin}$ is the linear
matter power spectrum, evaluated here at redshift $z_{*}$. We see that the
galaxy-galaxy lensing angular power spectrum depends on: 1) the density
fluctuations at $z_{*}$ where GR is recovered; 2) the evolution of background
quantities, $\mathcal{H}(z)$ and $\chi(z)$, which we assume here to be as in
$\Lambda$CDM 222This is an assumption that is often done in large-scale
structure analyses, since all current constraints set the background evolution
to be close to that of $\Lambda$CDM. This assumption can be relaxed, but it
requires adding other data sets in the analysis (e.g. supernovae or BAO data)
that would constrain the evolution of these functions. This is, however,
beyond the scope of this analysis.; and 3) the functions $\hat{J}$ and
$\hat{b}_{i}$ defined as

$\displaystyle\hat{J}(z)$
$\displaystyle\equiv\frac{J(z)\sigma_{8}(z)}{D_{1}(z)}=\frac{J(z)\sigma_{8}(z_{*})}{D_{1}(z_{*})}\,,$
(4) $\displaystyle\hat{b}_{i}(z)$ $\displaystyle\equiv
b_{i}(z)\sigma_{8}(z)\,,$ (5)

where $b_{i}$ is the linear galaxy bias in the tomographic redshift bin $i$.
The second equality in Eq. (4) follows from the fact that $\sigma_{8}(z)$ is
directly proportional to $D_{1}(z)$. As a consequence $\hat{J}$ and $J$
contain the same information about modified gravity, since they are related by
the ratio $\sigma_{8}(z_{*})/D_{1}(z_{*})$ which does not depend on the
modifications of gravity (since we assume that at $z_{*}$ GR is recovered).

Since $\hat{b}_{i}(z)$ and $\hat{J}(z)$ vary slowly with redshift, we can take
them out of the integral in Eq. (3) and evaluate them at the mean redshift of
the bin $z_{i}$. We see therefore, that by measuring the galaxy-galaxy lensing
angular power spectrum for lenses at redshift $z_{i}$, we can directly measure
the functions $\hat{J}$ and $\hat{b}_{i}$ at that redshift, in a fully model-
independent way, i.e. without assuming any theory of gravity, nor specifying a
redshift evolution for $\hat{J}$. The fact that galaxy-galaxy lensing depends
on $\hat{J}$ at the redshift of the lenses follows from the fact that, in the
Limber approximation, the signal is fully due to correlations between the Weyl
potential and the galaxy density at the position of the lenses. All the other
correlations along the photon’s trajectory are negligible. The situation for
the shear angular power spectrum is different, since it depends on the
integral of $\hat{J}$ from the sources to the observer. For this reason we do
not consider it in our analysis. Note however that this could be done by
introducing a function $\hat{J}$ which is piece-wise constant in some chosen
redshift bins, with smooth interpolation between them.

Equation (3) shows that there is a degeneracy between $\hat{J}$ and
$\hat{b}_{i}$, but this degeneracy can be easily broken by adding the
clustering angular power spectrum which only depends on $\hat{b}_{i}$.
Following DES:2021bpo , we use the Limber approximation to compute the galaxy
clustering spectrum at large $\ell$

$\displaystyle C_{\ell}^{\Delta\Delta}(z_{i},z_{j})=$
$\displaystyle\int\text{d}z\,n_{i}(z)n_{j}(z)\frac{\mathcal{H}(z)(1+z)}{\chi^{2}(z)}\hat{b}_{i}(z)\hat{b}_{j}(z)B\left(k_{\ell},\chi\right)\frac{P_{\delta\delta}^{\rm
lin}\left(k_{\ell},z_{*}\right)}{\sigma_{8}^{2}(z_{*})}\,,\quad\mbox{for
$\ell\geq 200$}\,.$ (6)

For low $\ell$, however, the Limber approximation is not accurate enough and
we need to go beyond it. Following Fang:2019xat , we split the non-linear
matter power spectrum into its linear contribution and the difference between
the non-linear and linear contributions: $P_{\delta\delta}^{\rm
nl}=P_{\delta\delta}^{\rm lin}+(P_{\delta\delta}^{\rm
nl}-P_{\delta\delta}^{\rm lin})$. For the purely non-linear part,
$P_{\delta\delta}^{\rm nl}-P_{\delta\delta}^{\rm lin}$, the Limber
approximation can be used, since non-linearities will only contribute at small
scales (large $k$) where the Limber approximation is valid. The linear part on
the other hand is computed exactly for $\ell<200$. In our formalism we can
write it as (this corresponds to the second term of Eq. (2.10) in Fang:2019xat
)

$\displaystyle\left.C_{\ell}^{\Delta\Delta}(z_{i},z_{j})\right|_{\rm lin}=$
$\displaystyle\frac{2}{\pi}\int\text{d}\chi_{1}\,n_{i}(\chi_{1})(1+z(\chi_{1}))\mathcal{H}(\chi_{1})\hat{b}_{i}(\chi_{1})\int\text{d}\chi_{2}\,n_{j}(\chi_{2})(1+z(\chi_{2}))\mathcal{H}(\chi_{2})\hat{b}_{j}(\chi_{2})$
$\displaystyle\times\int_{0}^{\infty}\text{d}k\,k^{2}\frac{P_{\delta\delta}^{\rm
lin}(k,z_{*})}{\sigma^{2}_{8}(z_{*})}j_{\ell}(k\chi_{1})j_{\ell}(k\chi_{2})\,,\quad\mbox{for
$\ell<200$}\,.$ (7)

Equations (6) and (II) are similar to Eq. (3): they depend on the density
power spectrum at $z_{*}$, on the background evolution through
$\mathcal{H}(z)$ and $\chi(z)$, and on the bias of the lens sample
$\hat{b}_{i}$. Combining galaxy clustering with galaxy-galaxy lensing allows
us therefore to measure separately $\hat{b}_{i}$ and $\hat{J}$.

Equations (3) and (6) depend on the boost $B(k,z)$ encoding the evolution of
matter density fluctuations in the non-linear regime. Without specifying a
theory of gravity, we cannot know the form of the boost. Here we follow the
strategy presented in DES:2022ygi  for extensions beyond GR, and assume that
the boost is the same as in GR. We then restrict the range of angular scales
used in the analysis to minimise the impact of the boost. We consider two
scenarios: one where we use all scales that were used in the standard DES
analysis DES:2021wwk , and another one where we impose a more stringent scale
cut similar to the one used in DES:2022ygi . Note that even though choosing a
boost as in GR is obviously not a model-independent treatment, it does not
invalidate our analysis. Suppose that we observe $\hat{J}$ different from GR
using our formalism. This indicates without ambiguity that GR is invalid (if
GR would be valid, the boost would be correct). One can then compare the
measured $\hat{J}$ with modified gravity models and find those that are
compatible with the measurements. Within these models, one can then rerun the
analysis, using now the correct boost in those models to refine the
constraints (which we do not expect to change by much, given that we have
chosen stringent angular scale cuts).

##  III Methods

To measure the Weyl potential with the DES observations, we follow closely the
baseline analysis for galaxy clustering and galaxy-galaxy lensing presented in
DES:2021bpo , which is also called $2\times 2$pt analysis. This includes the
data sets presented in that analysis, as well as the modelling of systematic
effects and the modelling of the cosmological observables. In the following we
describe the main differences with respect to the baseline analysis considered
in DES:2021bpo  and publicly-available in the CosmoSIS
333https://cosmosis.readthedocs.io/en/latest/ software Zuntz:2014csq .

Starting with the data sets, we consider the magnitude-limited MagLim sample
for the lenses, and the source sample obtained with Metacalibration. The full
lens sample contains 6 tomographic bins, which would in principle allow us to
determine $\hat{J}$ at 6 different effective redshifts. However, some residual
systematic uncertainties were identified in the two bins with the highest
redshift and this led to the use of only the first 4 bins in the baseline
$3\times 2$pt analysis DES:2021wwk . In this analysis we use a conservative
approach and focus only on the first 4 tomographic bins, even if we only
consider a $2\times 2$pt analysis. The mean redshift of the 4 tomographic bins
are: [0.295, 0.467, 0.626, 0.771].

In terms of modelling the (observational and instrumental) systematic effects,
we include nuisance parameters for the width and the position of the
distributions of the lenses, as well as nuisance parameters for the shift of
the source distributions. We use the officially-provided informative priors on
the corresponding nuisance parameters. We further consider the effect of
intrinsic alignments in our galaxy-galaxy lensing observables. Although the
baseline adopted in the $3\times 2$pt analysis by the DES Collaboration was
the tidal alignment and tidal torquing model Blazek:2017wbz , we follow the
approach used by the Collaboration for extended models and consider the
simpler non-linear alignment model Bridle:2007ft . Given the lack of
understanding of intrinsic alignments in modified gravity scenarios, we prefer
to take a conservative approach and treat these effects as if we were in GR.
The main reason behind this choice is to avoid artificially extracting
information from modified gravity from a wrong modelling of intrinsic
alignments. We further marginalise over the amplitude and redshift dependence
of intrinsic alignments to avoid possible biases in our posteriors due to the
intrinsic alignment assumptions.

Similar to the treatment of the intrinsic alignment of galaxies, we account
for magnification effects in the lens sample assuming GR. In principle,
magnification could be written in terms of the function $\hat{J}$, thus
providing further constraints on it. However, since this effect is strongly
subdominant, we expect the improvement to be negligible and we treat therefore
magnification as a nuisance, that can be computed in GR without biasing the
results. We fix the magnification bias parameters, as it was done in
DES:2021bpo . We also include the effect of RSD in the galaxy clustering
observable, and as for magnification we model it in GR, which again is well
justified since the effect is strongly subdominant. Following DES:2021bpo  we
consider a linear galaxy bias model, as already mentioned in Sect. II. We
consider broad flat priors for these parameters, given the fact that
$\hat{b}_{i}$ now contains information relative to the growth, see Eq. (5).
Finally, we account for the shear multiplicative bias using the publicly-
available informative priors. Concerning the galaxy-galaxy lensing observable,
we further marginalise analytically over the mass enclosed below the angular
scales considered in the analysis, as was done in DES:2021bpo . Contrary to
the baseline analysis, and in order to avoid dealing with the small scales, we
prefer not to include any information from shear ratios in the likelihood.

We have considered the MultiNest 444https://github.com/farhanferoz/MultiNest
Handley:2015fda  sampler to explore the parameter space with a tolerance of
0.1, 500 live points, and an efficiency of 0.3. However, we have cross-checked
the baseline results using PolyChord 555https://github.com/PolyChord/
Feroz:2008xx  with 500 live points, number of repetitions of 60, tolerance of
0.01, and boost of the posteriors of 10. The agreement on the mean values of
the parameters from the 2 samplers for our baseline scenario is excellent,
better than 0.5%. The errors on the parameters however increase slightly using
PolyChord, by $5-12\%$.

In order to test the dependency of our results on the different assumptions,
we consider four different scenarios in this analysis. Our baseline
corresponds to the use of $3\sigma$ Gaussian Planck priors on the cosmological
parameters Planck:2018vyg , while using the fiducial DES angular scale cuts.
In this way, the density power spectrum at high redshift $z_{*}$ (which we
choose to be $z_{*}=10$, when GR is recovered) is constrained by Planck, while
the late time evolution is left free and can differ from GR. Since Planck
constraints are obtained assuming the validity of $\Lambda$CDM we do not want
to introduce information from Planck on late time cosmology. Therefore we
consider the priors obtained from temperature and polarisation, but without
lensing. To test the sensitivity of our treatment on the boost factor for the
non-linear correction, we also consider a scenario with $3\sigma$ Planck
priors but using a more stringent angular scale cut, for which only scales
above $21\,$Mpc$/h$ are used. Although the method to apply the scale cut is
slightly different from the one considered in DES:2022ygi , the remaining data
points in the data vector are almost the same. In this pessimistic scenario,
essentially only linear scales are considered in the analysis. Finally, we
also consider two scenarios (with standard and pessimistic scale cuts),
removing the priors from Planck. In these scenarios, both the early time power
spectrum and the late time evolution of the Weyl potential are constrained by
DES data only. For these cases we use the wide fiducial DES priors on the
cosmological parameters.

To analyse the different chains and visualise posteriors, we use the GetDist
package Lewis:2019xzd .

##  IV Results

Figure 1: Measured values of $\hat{J}$ at the mean redshifts of the MagLim sample. The left panel shows the measurements assuming a prior from Planck on early time cosmological parameters. The right panel shows the measurements without any priors. The black and orange solid lines show the predictions for $\hat{J}$ in $\Lambda$CDM. The shaded regions indicate the 1$\sigma$ uncertainty on those predictions due to uncertainties in the primordial cosmological parameters. Table 1: Mean values and 1$\sigma$ error bars for $\hat{J}(z_{i})$ in the four cases considered in our analysis: with and without Planck priors, with standard and pessimistic scale cuts. | CMB prior | No prior
---|---|---
|  Standard cuts |  Pessimistic cuts |  Standard cuts |  Pessimistic cuts
$\hat{J}(z_{1})$ | $0.326\pm 0.013$ | $0.332\pm 0.025$ | $0.327\pm 0.016$ | $0.340\pm 0.027$
$\hat{J}(z_{2})$ | $0.331\pm 0.017$ | $0.326\pm 0.030$ | $0.327\pm 0.019$ | $0.331\pm 0.032$
$\hat{J}(z_{3})$ | $0.387\pm 0.025$ | $0.426\pm 0.057$ | $0.389\pm 0.029$ | $0.439\pm 0.058$
$\hat{J}(z_{4})$ | $0.354\pm 0.033$ | $0.361\pm 0.068$ | $0.347^{+0.036}_{-0.040}$ | $0.374\pm 0.071$

###  IV.1 Measurements of $\hat{J}$ using priors from Planck

In Fig. 1, left panel, we show the measured values of $\hat{J}$ with 1$\sigma$
error bars. We see that $\hat{J}$ is very well measured, with a precision of 4
to 9% in the four DES redshift bins (see also Table 1). This demonstrates the
capacity of galaxy-galaxy lensing and galaxy clustering to significantly
constrain the Weyl evolution in a fully model-independent way, i.e. without
assuming any theory of gravity at late time. The measured values of $\hat{J}$
can then be compared with any model of interest. In Fig. 1, we also show the
prediction for $\hat{J}$ in $\Lambda$CDM, with the cosmological parameters
obtained through our cosmological analysis. The shaded region corresponds to
the 1$\sigma$ uncertainty on the $\Lambda$CDM prediction due to the
uncertainty in the measurement of the cosmological parameters. In the first
and second redshift bins, the measurements of $\hat{J}$ are 2.3$\sigma$,
respectively 3.1$\sigma$, below the $\Lambda$CDM predictions. In the highest
two bins the measurements are compatible with $\Lambda$CDM (1.7$\sigma$
difference in the last bin).

Even though such a tension is not very significant, especially given that the
measurements of $\hat{J}$ in the different bins are slightly correlated (see
the joint constraints in Fig. 6, Appendix C), it is interesting to see at
which redshifts the measurements are in tension with $\Lambda$CDM. In
particular, Fig. 1 shows that the deviations from $\Lambda$CDM are localised
at the first two redshift bins, and actually more prominent in the second one.
This could hint either at a modification of gravity that may have more impact
at low redshift, when the mechanism responsible for the acceleration of the
Universe is in full play. Or it could be due to a (partially) incorrect
modelling of non-linear effects, which are more important at low redshift.
Alternatively, it could also be due to unknown systematic effects in the first
two bins of the MagLim sample. Since there is an important difference between
the measurements in the second and third bins, it would be interesting to
obtain a measurement in between.

Until now, $\hat{J}$ has been measured in a fully model-independent way. It is
however interesting to use these measurements to infer $\sigma_{8}(z=0)$
_assuming_ $\Lambda$CDM. This value of $\sigma_{8}(z=0)$, which is inferred
from the Weyl potential at low redshift, can then be compared with the value
obtained from high redshift, i.e. from the cosmological parameters
$\Omega_{\rm m},\Omega_{\rm b},n_{\rm s}$, and $A_{\rm s}$ that are
constrained through the density power spectrum at $z_{*}=10$, the background
functions $\mathcal{H}(z)$ and $\chi(z)$, and through Planck priors 666Note
that the errors on all parameters are dominated by Planck priors, except for
$\Omega_{\rm m}$ whose errors are a factor 2 tighter, due to constraints from
the background evolution, see Table 3 in Appendix A.. In $\Lambda$CDM,
$\hat{J}$ is given by

$\displaystyle\hat{J}(z)=\Omega_{\rm
m}(z)\frac{D_{1}(z)}{D_{1}(z=0)}\sigma_{8}(z=0)\,.$ (8)

Combining the four measurements of $\hat{J}$, we can fit for $\sigma_{8}(z=0)$
at each point of the chain, and obtain its posterior distribution which we
find to be close to Gaussian (see Fig. 7 in Appendix C). The mean and
1$\sigma$ errors are $\sigma^{\rm fit}_{8}(z=0)=0.74\pm 0.04$. In contrast,
using the values of $\Omega_{\rm m},\Omega_{\rm b},n_{\rm s}$ and $A_{\rm s}$,
we obtain $\sigma^{\rm cosmo}_{8}(z=0)=0.85\pm 0.03$, which is 2.5$\sigma$
higher than the value obtained from $\hat{J}$. With our method, we recover
therefore the well-known result from the Year 3 DES data that high redshifts
prefer a larger clustering amplitude than low redshifts. Moreover, we are able
to pin-point this (mild) tension to the behaviour of the Weyl potential in the
two lowest redshift bins of DES. Interestingly, the value of $\sigma_{8}(z=0)$
obtained from the early time cosmological parameters is slightly larger (by
1.3$\sigma$) in our analysis than when only Planck data are used: $0.85\pm
0.03$ versus $0.810\pm 0.007$. As we will see below, this tendency becomes
even more prominent when no priors are imposed on the primordial cosmological
parameters. Note that we obtain similar results if we use $\hat{J}$ to infer
$S_{8}=\sigma_{8}\sqrt{\Omega_{\rm m}/0.3}$. In this case, the tension between
the late time measurement of $S_{8}$ and the value obtained from early time
cosmological parameters is 2.4$\sigma$.

In this analysis we have used priors on cosmological parameters from Planck
coming from the temperature and the polarisation, but without lensing, since
Planck constraints from lensing assume the validity of $\Lambda$CDM at late
time. However, even if lensing is not included as an additional observable, it
still affects the temperature power spectrum. One way to remove this
information is to consider Planck analysis with an additional parameter
$A_{\rm L}$ in front of the amplitude of the lensing contribution, and to
marginalise over it. This effectively removes the constraining power of
lensing on cosmological parameters, ensuring that those are only constrained
by early time fluctuations. As a cross-check, we have redone our analysis,
using these new priors from Planck. We find that the results barely change.
The tension between the measured $\hat{J}$ and the $\Lambda$CDM prediction
slightly decreases (e.g. from 3.1$\sigma$ to 2.7$\sigma$ in the second bin)
due to the fact that Planck priors increase with this extra parameter $A_{\rm
L}$.

Finally, we have redone our analysis using the sampler PolyChord instead of
MultiNest. The mean values of $\hat{J}$ change by less than 0.5%. The errors
however increase slightly, by $7-12\%$. The errors on cosmological parameters
also increase by a similar amount ($5-9\%$), leading to a tension with
$\Lambda$CDM predictions which is very slightly reduced: 2$\sigma$ in the
first bin and 2.7$\sigma$ in the second bin, compared to 2.3$\sigma$ and
3.1$\sigma$ when using MultiNest.

Table 2: The values for $\sigma_{8}(z=0)$, in the cases with and without CMB priors, using the standard scale cuts. We compare the value fitted from $\hat{J}$ with the value obtained from the cosmological parameters. | CMB prior | No prior
---|---|---
| From parameters | From $\hat{J}$ | From parameters | From $\hat{J}$
$\sigma_{8}(z=0)$ |  $0.852\pm 0.027$ |  $0.741\pm 0.035$ |  $1.037\pm 0.093$ |  $0.780^{+0.064}_{-0.073}$

###  IV.2 Measurements of $\hat{J}$ without Planck prior

In the right panel of Fig. 1, we show the measured values of $\hat{J}$ when
the cosmological parameters are left completely free to vary, i.e. without
imposing a prior from Planck. We see that $\hat{J}$ can still be very well
measured in this case: the errors increase by $\sim 20\%$ only (see Table 1).
On the other hand, the errors on the cosmological parameters are 3 to 6 times
larger when no priors are used. The primordial amplitude $A_{\rm s}$ and the
Hubble parameter $h$ are the parameters that are the most degraded by removing
the priors. This is not surprising, since in our framework cosmological
parameters are only constrained by the primordial density power spectrum at
$z_{*}=10$ and by the evolution of the background, namely $\mathcal{H}(z)$ and
the comoving distance $\chi(z)$. All the information coming from the growth of
structure and from the Weyl potential is encoded in the parameters $\hat{J}$
and $\hat{b}_{i}$ and does not contribute to the constraints on the
cosmological parameters. The whole goal of our approach is indeed to assume
that we do not know how the Weyl potential evolves at late time. The larger
uncertainties in cosmological parameters lead to wider errors on the
prediction for $\hat{J}$ in $\Lambda$CDM, as is clearly visible from Fig. 1.

We see that removing the priors actually slightly increases the tension
between the measured $\hat{J}$ and the $\Lambda$CDM predictions, which is now
$[2.3\sigma,3.1\sigma,1.9\sigma,2.6\sigma$] in the four bins. From Fig. 1, we
see that this is due to the fact that the $\Lambda$CDM predictions for
$\hat{J}$ increase when the prior is removed. This can be traced to an
increase of the primordial amplitude $A_{\rm s}$ from $(2.14\pm 0.09)\times
10^{-9}$ to $3.04^{+0.55}_{-0.68}\times 10^{-9}$. Hence we see that DES data
_on their own_ prefer a high amplitude of perturbations at high redshift,
followed by a growth of the Weyl potential slower than in $\Lambda$CDM at low
redshifts. Again, this could hint to a modification of gravity that would slow
down the growth of the Weyl potential at late time. More precisely,
modifications of gravity can change the relation between density perturbations
at $z_{*}=10$ which affect the lensing signal through
$P_{\delta\delta}(z_{*})$, see Eq. (3), and the evolution of the Weyl
potential at low redshift encoded in $\hat{J}$. The tension could also be due
to an imperfect modelling of non-linearities, that would also affect the
relation between the linear density power spectrum at $z_{*}$ and the
evolution of the Weyl potential. Alternatively, it could be due to systematic
effects. In any case, it is highly interesting that the tension remains when
CMB data are not used at all.

The tension in $\hat{J}$ leads to a 2.2$\sigma$ tension between $\sigma_{8}$
extracted from the cosmological parameters: $\sigma^{\rm
cosmo}_{8}(z=0)=1.04\pm 0.09$, and $\sigma_{8}$ extracted from $\hat{J}$
assuming $\Lambda$CDM: $\sigma^{\rm fit}_{8}(z=0)=0.78\pm 0.07$. The very
large value of $\sigma^{\rm cosmo}_{8}$ is directly related to the large value
of $A_{\rm s}$. The $\sigma_{8}$ tension is sometimes referred to as a tension
between CMB data and weak lensing data. Here, we see that it is truly a
tension between low and high redshift that is encoded into DES data on their
own. Translating the constraints on $\sigma_{8}(z=0)$ to constraints on
$S_{8}$ we find that the tension is slightly larger, of 2.7$\sigma$. We
summarise the values of $\sigma_{8}(z=0)$ obtained in various cases in Table
2.

Finally, we have looked at the degeneracies of $\hat{J}$ with the nuisance
parameters. We find that $\hat{J}$ is degenerated with the stretch of the
photometric redshift distribution of the lens sample (see Fig. 8 in Appendix
C). This stretch is added as a free parameter in the baseline DES analysis,
together with a shift, to account for photometric redshift systematic effects
DES:2021bpo . The stretch and the shift are calibrated using clustering with
spectroscopic surveys DES:2020sjz , leading to tight priors on these
parameters. We find that a large value of the stretch, i.e. a distribution
which is more spread, can be counterbalanced by smaller values of $\hat{J}$,
and vice-versa. This can be understood from Eq. (3): increasing the width of
$n_{i}(z)$ tends to increase the values of the integral over $z$, due to the
lensing kernel
$(\chi(z)-\chi^{\prime}(z^{\prime}))/(\chi(z)\chi^{\prime}(z^{\prime}))$,
which increases when $\chi(z)$ is allowed to take smaller values. In order to
have $\hat{J}$ in 1$\sigma$ agreement with $\Lambda$CDM predictions in the
second bin, we would need a stretch of 0.94, which is in clear tension with
the values obtained from calibration: $1.306\pm 0.093$ (the mean value from
our analysis is $1.239\pm 0.076$). Table 3 in Appendix A summarises the mean
values and error bars for all nuisance parameters.

Figure 2: Measured values of $\hat{J}$ at the mean redshifts of the MagLim
sample, using more stringent scale cuts. The left panel shows the measurements
assuming a prior from Planck on early time cosmological parameters. The right
panel shows the measurements without any priors. The black and orange solid
lines show the predictions for $\hat{J}$ in $\Lambda$CDM. The shaded regions
indicate the 1$\sigma$ uncertainty on those predictions due to uncertainties
in the primordial cosmological parameters.

###  IV.3 Measurements of $\hat{J}$ with more stringent non-linear scale cuts

As discussed in Section III, we have implemented more stringent scale cuts to
remove further the impact of non-linear scales. The results for $\hat{J}$ are
shown in Fig. 2 for the analysis with Planck priors (left panel) and without
(right panel). We see that the measurements of $\hat{J}$ are significantly
degraded: the errors increase by roughly a factor 2. This is not surprising
since removing scales reduces the available amount of information needed to
measure $\hat{J}$. The errors on the $\Lambda$CDM prediction also increase due
to the scale cut, in the case without prior, but only by 30%. Interestingly,
the mean values of $\hat{J}$ are almost not affected by removing scales and
are still below the $\Lambda$CDM predictions. However, the increase in the
error bars significantly reduces the tension with $\Lambda$CDM predictions:
only the second bin is still 2$\sigma$ away from the $\Lambda$CDM prediction
in the case with prior (1.6$\sigma$ without prior). A comparison of the
measurements of $\hat{J}$ with standard and pessimistic scale cuts is provided
in Table 1.

###  IV.4 Comparison with binned $\sigma_{8}(z)$ analysis of DES

It is worth comparing our results with the binned $\sigma_{8}(z)$ analysis of
DES DES:2022ygi . In this analysis, $\Lambda$CDM is assumed to be correct, and
four additional free parameters, $A_{i}^{P_{\rm lin}}$, are added in front of
the density power spectrum in the four redshift bins of the lenses. The
parameter in the first bin is then fixed to 1 (to break the degeneracy with
the primordial amplitude $A_{\rm s}$), whereas the other three are constrained
by the data. This analysis differs from ours in the sense that it is a
consistency check of $\Lambda$CDM. If the parameters are found to be different
from 1 in any of the bins, then $\Lambda$CDM is inconsistent. However, this
analysis does not permit a determination of the origin of the tension. In
particular, since $\Lambda$CDM is assumed to be correct, the Weyl potential is
related to the matter density fluctuation using GR. As a consequence, the
angular power spectra are all sensitive to the matter power spectrum, which is
multiplied by a unique parameter $A_{i}^{P_{\rm lin}}$ (per bin). In reality
however, the clustering angular power spectrum depends on the density-density
correlation; the galaxy-galaxy lensing angular power spectrum depends on the
density-Weyl correlation; and the shear angular power spectrum depends on the
Weyl-Weyl correlation. Modifications of gravity are not expected to modify the
evolution of the density and the evolution of the Weyl potential in the same
way. Therefore, encoding those into a single parameter may not be optimal. Of
course, as explained in DES:2022ygi , the goal of such a consistency check is
not to model modified gravity, but rather to assess the consistency of
$\Lambda$CDM. However, the fact that one has only one parameter per bin
introduces an artificial mixing between this parameter and the galaxy bias,
which could in principle reduce the signature of modified gravity. Deviations
from GR in the Weyl potential would indeed, in this framework, lead to
modifications of the parameters $A_{i}^{P_{\rm lin}}$, which would
artificially change the clustering power spectrum as well (since
$A_{i}^{P_{\rm lin}}$ multiplies all spectra). If the matter density is not
modified, or if it is modified in a different way than the Weyl potential,
this will lead to inconsistencies in the modelling that may be minimised by
artifically decreasing deviations in $A_{i}^{P_{\rm lin}}$ and changing the
bias or other cosmological parameters. In contrast, with our modelling, any
deviation in the Weyl potential would be automatically visible in the
measurement of $\hat{J}$, while deviations in the growth of the density would
affect the parameters $\hat{b}_{i}(z)=b_{i}(z)\sigma_{8}(z)$.

A direct comparison of our results with the binned $\sigma_{8}(z)$ results is
not possible, since the assumptions are different. However, from Fig. 11 and
12 of DES:2022ygi , one can see that when using only DES data, $\sigma_{8}$
tends to be lower than predicted by Planck using $\Lambda$CDM, at all
redshifts. Combining with Planck increases the values, bringing them back
towards the $\Lambda$CDM predictions. In our case, we find that starting with
a power spectrum consistent with Planck at high redshift, and letting the
evolution free at low redshift, leads to a Weyl evolution below $\Lambda$CDM
predictions. As for the binned $\sigma_{8}(z)$ analysis, this trend is not
very significant. However, the main difference is that in the second bin we
have a value which is 3.1$\sigma$ below the $\Lambda$CDM prediction. This is
not in contradiction with the results of the binned $\sigma_{8}(z)$ analysis,
which does not see such a feature when DES and Planck are combined. By
assuming $\Lambda$CDM to be valid and fixing $A_{1}^{P_{\rm lin}}=1$ in the
lowest bin, the constraints from different bins are not truly independent. In
particular, a deviation in one bin may lead to slightly different values of
the $\Lambda$CDM parameters, which in turn will artificially influence the
value of $A_{i}^{P_{\rm lin}}$ in the other bins. Moreover, the linear galaxy
bias can also reabsorb part of the deviation. In our framework this is not the
case: a deviation in one redshift bin does not influence the measurements in
the other bins. The correlations between $\hat{J}$ in different bins is purely
physical, only due to the evolution of the Weyl potential. From this we see
that identifying the quantities that are truly measured by the data helps to
correctly pin-point where tensions come from.

###  IV.5 Measurements of $\hat{J}$ using the redMaGiC lens sample

In our analysis we have used the MagLim lens sample, which is the fiducial
sample of DES. It is, however, interesting to see how the results change if we
use the redMaGiC sample instead. For our baseline scenario (with Planck priors
and standard scale cuts), we find that the measured $\hat{J}$ using redMaGiC
are systematically below the $\Lambda$CDM predictions (see Fig. 5 and Table 4
in Appendix B). The tension with $\Lambda$CDM in the five bins of redMaGiC are
$[2.9,3.2,2.6,2.3,3.1]\sigma$. Starting from these measurements and assuming
the validity of $\Lambda$CDM, we find that the inferred value for $\sigma_{8}$
at late time is $\sigma^{\rm fit}_{8}(z=0)=0.63\pm 0.04$, which is 4.1$\sigma$
below the value inferred from early time cosmological parameters: $\sigma^{\rm
cosmo}_{8}(z=0)=0.82\pm 0.03$.

This non-negligible tension with respect to $\Lambda$CDM predictions is in
agreement with the results of the DES Collaboration, who found that
$\sigma_{8}$ measured from galaxy-galaxy lensing and galaxy clustering is
significantly lower using redMaGiC than MagLim DES:2021zxv . To account for
this, an artificial parameter $X_{\rm lens}$ was introduced, encoding a
difference in the galaxy bias obtained from galaxy clustering and the bias
obtained from galaxy-galaxy lensing. This parameter was found to be in
5$\sigma$ tension with its expected value of 1, reflecting the fact that the
galaxy clustering amplitude was systematically larger than the galaxy-galaxy
lensing amplitude DES:2021zxv ; DES:2021wwk . This is fully in line with our
findings: leaving the function $\hat{J}$ free, we find that the data prefer a
low value of $\hat{J}$ consistent with a galaxy-galaxy lensing correlation
lower than in $\Lambda$CDM. The DES Collaboration relates the low value of
$X_{\rm lens}$ to systematic effects due to the selection of galaxies in the
redMaGiC sample. Such systematic effects would bias our measurements of
$\hat{J}$, which is degenerated with the galaxy bias in the galaxy-galaxy
lensing signal. One way to investigate this would be to measure $\hat{J}$ from
shear-shear correlations and see if the results are consistent. We defer this
to a future analysis.

##  V Modified gravity

Figure 3: Predictions for $\hat{J}$ for modified gravity models with
$\Sigma\neq 1$ and $\mu=1$. We show the mean value together with the 1$\sigma$
uncertainty for the three choices of time evolution (standard, constant, and
exponential) described in the text. The measurements correspond to our
baseline, using Planck priors and with standard scale cuts.

The measurements of $\hat{J}$ are fully model-independent and they can
therefore be compared with predictions in any theory of gravity (as long as GR
is recovered at high redshift), without having to redo the analysis. Here, as
an illustration, we consider two cases: A) the phenomenological $\mu-\Sigma$
extensions of gravity; and B) Horndeski theories, concentrating on a few
specific cases.

###  V.1 $\mu-\Sigma$ modifications of gravity

The phenomenological parameters $\mu$ and $\Sigma$ encode deviations from GR
in the relation between the metric potentials and the matter density
fluctuations,

$\displaystyle k^{2}\Psi=-4\pi\mu(a,k)Ga^{2}\rho\delta\,\quad\mbox{and}\quad
k^{2}\Psi_{W}=-4\pi\Sigma(a,k)Ga^{2}\rho\delta\ .$ (9)

In GR, $\mu=\Sigma=1$. These parameters affect $\hat{J}$ in the following way,

$\displaystyle\hat{J}(z)=\Sigma(z)\Omega_{\rm
m}(z)D_{1}(z)\frac{\sigma_{8}(z_{*})}{D_{1}(z_{*})}\,,$ (10)

where as before we have neglected the dependence in $k$, as is usually done in
the literature, see e.g. DES:2022ygi  (both due to the fact that current data
are not constraining enough to probe a possible scale-dependence in $\mu$ and
$\Sigma$, and that in many models of modified gravity the scale-dependence is
negligible for sub-horizon scales). $\hat{J}$ depends directly on $\Sigma$,
but it also depends on $\mu$ through the evolution of $D_{1}(z)$. Eq. (10)
underlines the difference between measuring $\hat{J}$, and measuring $\mu$ and
$\Sigma$. $\hat{J}$ can be measured from lensing data on their own. The
parameters $\mu$ and $\Sigma$, on the other hand, cannot be measured
separately from lensing only, since their impact on $\hat{J}$ is fully
degenerated. To determine if $\mu$ and $\Sigma$ are consistent with GR, it is
therefore necessary to include other measurements, like RSD. Moreover, in
order to constrain $\mu$ from RSD, one has to assume the validity of Euler’s
equation for dark matter. The growth of structure is indeed not only affected
by $\mu$, but also by additional forces or interactions affecting dark matter
SvevaNastassiaCamille ; Bonvin:2022tii . As a consequence, the constraints on
$\Sigma$ from lensing also rely on the validity of Euler’s equation. In
contrast, $\hat{J}$ can be measured without any assumptions on the behaviour
of dark matter. Finally, we see from Eq. (10) that $\hat{J}$ depends directly
on the redshift evolution of $\mu$ and $\Sigma$, which is unknown. In
particular, $\hat{J}$ depends on $\mu(z)$ through a second order evolution
equation, which requires knowledge of $\mu(z)$ not only at the redshift of the
analysis, but also at all redshifts above it.

In practice, we find that $\hat{J}$ is much more sensitive to $\Sigma$ than to
$\mu$. Here, we aim to investigate the constraining power of $\hat{J}$,
without the inclusion of additional RSD data. Therefore, as illustration, we
choose $\mu=1$ (as in GR) and we infer $\Sigma$ from $\hat{J}$. We consider 3
different choices of time evolution, encoded through

$\displaystyle\Sigma(z)=1+\Sigma_{0}g(z)\,,$ (11)

with 1) the standard evolution that has been used in the DES analysis
DES:2022ygi  and in Planck analysis Planck:2018vyg :
$g(z)=\Omega_{\Lambda}(z)$; 2) no evolution: $g(z)=1$ for $z\in[0,1]$ and 0
elsewhere; and 3) an exponential evolution: $g(z)=\exp(1+z)$ for $z\in[0,1]$
and 0 elsewhere. The first case is motivated by the fact that deviations from
GR are linked to the accelerated expansion of the Universe, and are therefore
expected to decay proportionally to the amount of dark energy. The second and
third cases are not physically motivated time evolutions, but they allow us to
explore the sensitivity of the data to various behaviours, drastically
different from each other.

At each point of the chain, we fit for $\Sigma_{0}$ using Eq. (10), combining
the four measurements of $\hat{J}$. The value $\sigma_{8}(z_{*})$ is computed
at each point from the cosmological parameters using CAMB
777https://camb.readthedocs.io/en/latest/ Lewis:1999bs . The posteriors for
$\Sigma_{0}$ are very close to Gaussian for the three cases, with mean and
1$\sigma$ errors given by: $\Sigma_{0}^{\rm st}=-0.24\pm 0.09$ (standard
evolution), $\Sigma_{0}^{\rm cst}=-0.14\pm 0.05$ (no evolution) and
$\Sigma_{0}^{\rm exp}=-0.03\pm 0.01$ (exponential evolution). In Fig. 3 we
show the predicted $\hat{J}(z)$ (mean and $1\sigma$ error bands) for the three
models, compared with the measured $\hat{J}$ in the baseline case (with Planck
priors and standard scale cuts). In the three cases, a negative value of
$\Sigma_{0}$ decreases $\hat{J}$ with respect to $\Lambda$CDM, leading to a
better agreement with the data (at the cost of one extra parameter though).
The reduced chi-squared of the three models is very similar: 1.6 for the
standard evolution and the constant case, and 1.5 for the exponential
evolution. This shows that the data are not able to discriminate between the
different time evolutions, as is clear from Fig. 3. The coming generation of
surveys has clearly the potential to help, by adding measurements at lower and
higher redshifts and reducing the error bars. Here we also see the strong
advantage of our method: by first measuring $\hat{J}$ without assuming any
model and then comparing with the predictions in modified gravity, we can
easily understand and test the impact of different assumptions about the time
evolution.

###  V.2 Horndeski theories and a non-minimal coupling of CDM

Figure 4: Best-fit and $1\sigma$ bands for a Horndeski model with $\alpha_{\rm
M}=2\alpha_{\rm B}$ (red) and a model with a non-minimal coupling,
$\gamma_{c}\neq 0$, of CDM to the metric (black). In both cases, all other
modified gravity parameters are assumed to maintain their GR values.

Besides these phenomenological modifications of GR, we also consider Horndeski
models Horndeski1974 , which constitute the most general class of Lorentz-
invariant scalar-tensor theories with second-order equations of motion. In the
regime of linear cosmological perturbations, this broad class of theories can
be expressed in an effective theory approach by a limited number of parameters
Bellini:2014fua , including the running of the Planck mass $\alpha_{\rm M}$
and the braiding $\alpha_{\rm B}$. Following the approach of Castello:2023zjr
; Gleyzes:2015pma ; Gleyzes:2015rua , we additionally allow for dark matter to
be non-minimally coupled to the scalar field, since the weak equivalence
principle has not been tested for dark matter. While theoretically easier to
interpret, since the modifications enter directly at the level of the
Lagrangian, Horndeski models with a non-minimal coupling of CDM add more
complexity compared to the phenomenological $\mu-\Sigma$ modifications. In
particular, they influence the background evolution of $\mathcal{H}(z)$ along
with the perturbations. As previously mentioned, relaxing the assumption of
the background evolution of $\Lambda$CDM would necessitate the inclusion of
further data sets, which is beyond the scope of this work. Instead, for
simplicity we neglect the impact on the background evolution and only
constrain the parameters through their impact on $\hat{J}$. As such, our
constraints should be seen as an illustration of the capability of this model
type to explain a $\hat{J}$ measurement different from the $\Lambda$CDM
prediction, rather than a full analysis of the models.

We focus on two specific cases: first, we consider the case where $\alpha_{\rm
B}=\alpha_{\rm M}/2$ can differ from zero, and dark matter is minimally
coupled to the scalar field, $\gamma_{c}=0$ (see e.g. Castello:2023zjr  for
more detail on these parameters). This special relation between $\alpha_{\rm
M}$ and $\alpha_{\rm B}$ is recovered e.g. in Brans-Dicke theories
Brans:1961sx ; Sawicki:2012re  and some $f(R)$ models 888We assume that the
effective equation of state parameter $\hat{w}_{\rm DE}$ maintains its GR
value of $-1$, see Section 5.1.3 of Castello:2023zjr  for more detail. This
choice of parameterisation ensures the stability of this specific model type.
Carroll:2003wy ; Song:2006ej ; Carroll:2006jn ; Vollick:2003aw . Second, we
investigate the case with a non-minimal coupling of dark matter,
$\gamma_{c}\neq 0$, while all other parameters are kept to their GR value. As
before, we need a time evolution for the parameters, which we choose to be
$g(z)=\Omega_{\Lambda}(z)$ as usually considered for Horndeski models. For
both cases, we consider the baseline analysis, with Planck priors and standard
scale cuts.

To find $\alpha_{\rm M,0}$ and $\gamma_{\rm c,0}$ we can use Eq. (10), which
is also valid in Horndeski theories. For each value of the parameters, we can
compute $\Sigma(z_{i})=\mu(z_{i})(1+\eta(z_{i}))$, using the relations between
the model parameters $(\alpha_{\rm M},\alpha_{\rm B},\gamma_{c})$ and the
phenomenological parameters $(\mu,\eta)$ stated in Appendix A of
Castello:2023zjr . We then use the EF-TIGRE code
999https://github.com/Mik3M4n/EF-TIGRE. developed in Castello:2023zjr  to
obtain $\Omega_{\rm m}(z_{i})$ and $D_{1}(z_{i})/D_{1}(z_{\ast})$. In addition
to their dependence on $\alpha_{\rm M,0}$ and $\gamma_{\rm c,0}$, these
quantities depend also on the primordial cosmological parameters. However,
since it is computationally expensive to solve the system of differential
equations at each point of the chain, we solve it instead for the mean values
of the primordial cosmological parameters obtained from our analysis.
Similarly, we compute the mean value of $\sigma_{8}(z_{*})$ using CAMB. With
this, we can compute $\hat{J}(z_{i})$ using Eq. (10). We then fit for
$\alpha_{\rm M,0}$, respectively for $\gamma_{c,0}$, using the mean values for
the four measured $\hat{J}$ and their covariance from our baseline analysis.
Since we do not fit for $\alpha_{\rm M,0}$ and $\gamma_{c,0}$ at each point of
the chain, we do not have posteriors for these quantities. We can, however,
compute the 1$\sigma$ error bars by propagating the errors on the primordial
parameters. In principle, the way in which the errors on $\Omega_{\rm m}$ and
$\Omega_{\rm b}$ at initial epoch propagate into the errors on $\Omega_{\rm
m}(z_{i})$ and $D_{1}(z_{i})/D_{1}(z_{\ast})$ depends on the values of the
modified gravity parameters. For simplicity, we assume that they are the same
as in $\Lambda$CDM.

For the two specific models, we obtain $\alpha_{\rm M,0}=2\alpha_{\rm
B,0}=0.54^{+0.19}_{-0.18}$ and $\gamma_{c,0}=-0.79^{+0.28}_{-0.33}$. We
illustrate our results in Fig. 4, which shows that both cases lead to a very
similar functional behaviour of $\hat{J}(z)$. We conclude that both
$\alpha_{\rm M}=2\alpha_{\rm B}>0$ as well as $\gamma_{c}<0$ can decrease the
amplitude of the Weyl potential at late time, leading to predicted values of
$\hat{J}(z)$ in line with the measured data. However, from measurements of
$\hat{J}(z)$ alone it is not possible to distinguish between modifications of
gravity encoded in $\alpha_{\rm M}$ and $\alpha_{\rm B}$, and a breaking of
the weak equivalence principle for CDM given a non-minimal coupling
$\gamma_{c}<0$. Nevertheless, the observable $\hat{J}$ provides an excellent
test of the $\Lambda$CDM model. A combination with other observables, in
particular gravitational redshift, will allow to break degeneracies between
different physical effects Castello:2023zjr .

##  VI Conclusion

In this paper, we have performed the first measurement of the Weyl potential
from galaxy-galaxy lensing. Using data from DES, we were able to measure the
function $\hat{J}(z)$, which encodes the evolution of the Weyl potential, with
a precision of 4 to 9% in the four bins of the MagLim lens sample. We found
that the measured values in the two lowest bins are in mild tension with the
$\Lambda$CDM predictions. Imposing more stringent scale cuts does not impact
the mean values of $\hat{J}$, which remain below the $\Lambda$CDM predictions.
However, it degrades the constraints by a factor of 2, thus decreasing the
tension.

One remarkable feature of our method is that it allows us to separate the
information coming from the matter density fluctuations (which we do not use
to constrain gravity, since it is fully degenerated with the galaxy bias) from
the information coming from the Weyl potential. The latter purely encodes the
growth of the spatial and temporal distortions of the geometry, and is
therefore independent of the galaxy bias. Moreover, our method does not rely
on any assumption about the redshift evolution of $\hat{J}$: the function is
directly measured in each bin of the survey.

Our measurements allow us to pin-point the $\sigma_{8}$ tension to the fact
that the Weyl potential is smaller than predicted in $\Lambda$CDM in the first
two bins of the DES data. Surprisingly, we found that the tension remains even
when no priors from the CMB are used. DES data on their own prefer a Universe
with a high initial value of the primordial potential (higher than measured by
Planck) followed by a slower growth at late time. Further investigation to
understand the origin of this behaviour is needed. In particular, it would be
interesting to redo the analysis changing the settings to the ones used in the
new combined analysis of DES and KiDS Kilo-DegreeSurvey:2023gfr  (which found
a decrease of the $\sigma_{8}$ tension) to see how this impacts the
measurements of $\hat{J}$. Moreover, having measurements of $\hat{J}$ between
the second and third bin of DES, where the deviation suddenly appears could
help understand the origin of the tension. Surveys like Euclid and LSST will
be of crucial importance for this.

More generally, our method allows us to compare the measurements of $\hat{J}$
with predictions in any theory of gravity that recovers GR at high redshift.
Since the measurements are model-independent, the method allows for various
tests of the theory of gravity in an efficient way. One can for example easily
change the time evolution of the parameters of the theory, without having to
reanalyse the data for each case. Using the $\mu-\Sigma$ parameterisation of
modified gravity, we have shown that current data are not constraining enough
to differentiate between different time evolutions. However, future surveys
like Euclid and LSST are expected to provide sufficiently tight constraints
(at the level of $0.1-1\%$ for $z\in[0.25,2.1]$ for LSST Tutusaus:2022cab )
over a wider redshift range, to allow for much more stringent tests.

In a forthcoming paper, we will study how the evolution of the Weyl potential
can be combined with measurements of the growth rate of structure, to test the
relation between these two fundamental quantities in our Universe.

Data availability statement

All the data used in this analysis have been produced and publicly released by
the DES Collaboration DES:2021wwk . It is accessible through the CosmoSIS
software Zuntz:2014csq  in https://cosmosis.readthedocs.io/en/latest/.

###### Acknowledgements.

We thank Agnès Ferté for useful comments. We acknowledge the use of the HPC
cluster Yggdrasil at the University of Geneva for conducting our analysis.
C.B. and N.G. acknowledge support from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (grant
agreement No. 863929; project title “Testing the law of gravity with novel
large-scale structure observables”). C.B. acknowledges support from the Swiss
National Science Foundation.

##  Appendix A List of cosmological and nuisance parameters

In Table 3, we list the mean values and 1$\sigma$ errors that we obtain for
all cosmological parameters and nuisance parameters. We show the results for
the baseline analysis, i.e. using CMB priors and standard scale cuts, and the
case where we use only DES data (also with standard scale cuts).

Table 3: Mean values and 1$\sigma$ errors for the cosmological parameters and nuisance parameters obtained from the analysis with and without CMB priors. The two cases are for standard scale cuts. |  |  CMB prior |  No prior
---|---|---|---
Matter density | $\Omega_{\rm m}$ | $0.328\pm 0.011$ | $0.292^{+0.027}_{-0.036}$
Baryon density | $\Omega_{\rm b}$ | $0.0490\pm 0.0020$ | $0.0424^{+0.0043}_{-0.011}$
Hubble parameter | $h$ | $0.679\pm 0.016$ | $0.728\pm 0.086$
Amplitude $\times 10^{9}$ | $A_{\rm s}$ | $2.137\pm 0.094$ | $3.04^{+0.55}_{-0.68}$
Spectral index | $n_{s}$ | $0.965\pm 0.012$ | $0.986^{+0.078}_{-0.032}$
Bias $z_{1}$ | $\hat{b}_{1}$ | $0.953\pm 0.033$ | $0.962\pm 0.041$
Bias $z_{2}$ | $\hat{b}_{2}$ | $1.022\pm 0.036$ | $1.059\pm 0.049$
Bias $z_{3}$ | $\hat{b}_{3}$ | $1.006^{+0.033}_{-0.029}$ | $1.038\pm 0.043$
Bias $z_{4}$ | $\hat{b}_{4}$ | $0.906\pm 0.024$ | $0.937\pm 0.038$
Shear calibration | $m^{1}$ | $-0.0060\pm 0.0085$ | $-0.0060\pm 0.0087$
Shear calibration | $m^{2}$ | $-0.0206\pm 0.0073$ | $-0.0203\pm 0.0074$
Shear calibration | $m^{3}$ | $-0.0244\pm 0.0070$ | $-0.0243\pm 0.0069$
Shear calibration | $m^{4}$ | $-0.0363\pm 0.0070$ | $-0.0365\pm 0.0071$
Intrinsic alignment | $A_{\rm IA}$ | $0.226^{+0.066}_{-0.086}$ | $0.264^{+0.077}_{-0.087}$
Intrinsic alignment | $\alpha_{\rm IA}$ | $-1.7^{+1.4}_{-2.4}$ | $-0.7^{+1.9}_{-2.4}$
Lens photo-$z$ shift $z_{1}$ | $\Delta z_{l}^{1}$ | $-0.0089\pm 0.0058$ | $-0.0094\pm 0.0060$
Lens photo-$z$ shift $z_{2}$ | $\Delta z_{l}^{2}$ | $-0.0269^{+0.0075}_{-0.0086}$ | $-0.0285\pm 0.0085$
Lens photo-$z$ shift $z_{3}$ | $\Delta z_{l}^{3}$ | $-0.0020\pm 0.0054$ | $-0.0013\pm 0.0055$
Lens photo-$z$ shift $z_{4}$ | $\Delta z_{l}^{4}$ | $-0.0067\pm 0.0056$ | $-0.0063\pm 0.0057$
Lens photo-$z$ stretch $z_{1}$ | $\sigma z_{l}^{1}$ | $0.978\pm 0.057$ | $0.978\pm 0.058$
Lens photo-$z$ stretch $z_{1}$ | $\sigma z_{l}^{1}$ | $1.211^{+0.071}_{-0.055}$ | $1.239\pm 0.076$
Lens photo-$z$ stretch $z_{1}$ | $\sigma z_{l}^{1}$ | $0.888\pm 0.048$ | $0.891\pm 0.050$
Lens photo-$z$ stretch $z_{1}$ | $\sigma z_{l}^{1}$ | $0.930\pm 0.044$ | $0.931\pm 0.044$
Source photo-$z$ shift $z_{1}$ | $\Delta z_{s}^{1}$ | $0.0098\pm 0.015$ | $0.0097\pm 0.015$
Source photo-$z$ shift $z_{2}$ | $\Delta z_{s}^{2}$ | $-0.0167^{+0.010}_{-0.0064}$ | $-0.0160^{+0.0096}_{-0.0073}$
Source photo-$z$ shift $z_{3}$ | $\Delta z_{s}^{3}$ | $-0.0190\pm 0.0076$ | $-0.0198\pm 0.0074$
Source photo-$z$ shift $z_{4}$ | $\Delta z_{s}^{4}$ | $0.008\pm 0.014$ | $0.009\pm 0.014$

##  Appendix B Results using the redMaGiC lens sample

Figure 5: Measured values of $\hat{J}$ at the mean redshifts of the redMaGiC
sample, in the baseline analysis, i.e. using priors from Planck on early time
cosmological parameters and standard scale cuts. The purple solid line shows
the prediction for $\hat{J}$ in $\Lambda$CDM. The shaded regions indicate the
1$\sigma$ uncertainty on this prediction due to uncertainties in the
primordial cosmological parameters.

In Fig. 5, we show the measurements of $\hat{J}$ in the five redshift bins of
the redMaGiC lens sample: $z\in[0.280,0.437,0.580,0.727,0.849]$, for our
baseline analysis, i.e. using Planck priors and standard scale cuts. We also
show the $\Lambda$CDM predictions using the cosmological parameters from our
analysis. The measurements are systematically below the predictions. The
values and 1$\sigma$ error-bars are listed in Table 4.

Table 4: Mean values and 1$\sigma$ error bars for $\hat{J}(z_{i})$ from the redMaGiC sample, in the baseline analysis. $\hat{J}(z_{1})$ | $0.291\pm 0.015$
---|---
$\hat{J}(z_{2})$ | $0.311\pm 0.016$
$\hat{J}(z_{3})$ | $0.328\pm 0.021$
$\hat{J}(z_{4})$ | $0.313\pm 0.034$
$\hat{J}(z_{5})$ | $0.241\pm 0.049$

##  Appendix C Additional plots

Figure 6: Joint contours for the four values of $\hat{J}$ in the baseline
scenario with Planck priors and standard scale cuts. Figure 7: Joint contours
for $\sigma^{\rm fit}_{8}(z=0)$ obtained from $\hat{J}$, and $\sigma_{8}^{\rm
cosmo}(z=0)$ obtained from cosmological parameters, in the baseline scenario
with Planck priors and standard scale cuts. Figure 8: Joint contours for
$\hat{J}$ and the shift $\Delta z_{l}$ and stretch $\sigma z_{l}$ of the
photometric distribution in the second bin of the MagLim sample, where the
tension is the largest. Here, we show the case without Planck prior and with
standard scale cuts.

Here, we show various joint distributions of parameters discussed in the main
text. In Fig. 6, we plot the joint contours for $\hat{J}$, showing that the
posteriors are close to Gaussian and that there is a mild correlation between
the different redshift bins. In Fig. 7, we show the joint distribution of
$\sigma_{8}(z=0)$ obtained from $\hat{J}$ and obtained from the cosmological
parameters. Finally, in Fig. 8 we show the joint distributions, in the second
redshift bin, of $\hat{J}$ and the two nuisance parameters $\Delta z_{l}$
(shift) and $\sigma z_{l}$ (width) that govern the photometric redshift
distribution of the lenses.

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