A non-linear solution to the S₈ tension – II. Analysis of DES Year 3 cosmic shear

ABSTRACT

Weak galaxy lensing surveys have consistently reported low values of the S₈ parameter compared to the Planck lambda cold dark matter (ΛCDM) cosmology. Amon & Efstathiou used KiDS-1000 cosmic shear measurements to propose that this tension can be reconciled if the matter fluctuation spectrum is suppressed more strongly on non-linear scales than assumed in state-of-the-art hydrodynamical simulations. In this paper, we investigate cosmic shear data from the Dark Energy Survey (DES) Year 3. The non-linear suppression of the matter power spectrum required to resolve the S₈ tension between DES and the Planck ΛCDM model is not as strong as inferred using KiDS data, but is still more extreme than predictions from recent numerical simulations. An alternative possibility is that non-standard dark matter contributes to the required suppression. We investigate the redshift and scale dependence of the suppression of the matter power spectrum. If our proposed explanation of the S₈ tension is correct, the required suppression must extend into the mildly non-linear regime to wavenumbers |$k\sim 0.2 \, h\, {\rm Mpc}^{-1}$|⁠. In addition, all measures of S₈ using linear scales should agree with the Planck ΛCDM cosmology, an expectation that will be testable to high precision in the near future.

1 INTRODUCTION

The six-parameter lambda cold dark matter (ΛCDM) cosmological model has been incredibly successful in explaining the anisotropies of the cosmic microwave background (CMB; e.g. Planck Collaboration 2020a), baryon acoustic oscillations (BAO; e.g. Alam et al. 2021), and a wide range of other astronomical data. Nevertheless, there are indications of ‘tensions’ between this model and some observations. The ‘Hubble tension’, i.e. the discrepancy between distance ladder measurements of the Hubble parameter at present day, H₀, and the value inferred from the CMB assuming the ΛCDM model, is the most well-known (for reviews see Freedman 2021; Shah, Lemos & Lahav 2021; Kamionkowski & Riess 2022). In addition, there is an apparent discrepancy between the amplitude of the matter fluctuations inferred from the CMB and that measured in cosmic shear surveys which has become known as the ‘S₈ tension’,¹ (e.g. Heymans et al. 2013; Asgari et al. 2021; Amon et al. 2022; Secco, Samuroff et al. 2022; Dalal et al. 2023; Li et al. 2023). Since new physics may be required to explain these tensions, they have become the focus of many recent observational and theoretical studies.

This paper is a sequel to Amon & Efstathiou (2022), hereafter Paper I, and is devoted exclusively to the S₈ tension. Paper I investigated the hypothesis that the Planck ΛCDM cosmology accurately describes matter fluctuations on linear scales, including their growth rate. The S₈ tension is then explained by modifying the matter power spectrum on non-linear scales. Paper I adopted a simple phenomenological model for the matter power spectrum

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

Our hypothesis predicts that all measures of the fluctuation amplitude dominated by linear scales, such as galaxy redshift-space distortions (e.g. Alam et al. 2021; D’Amico et al. 2022; Philcox & Ivanov 2022; Chen et al. 2022) and cross-correlations of CMB lensing with galaxy surveys (e.g. White et al. 2022; Chen et al. 2022; Chang et al. 2023) should agree with the predictions of the Planck ΛCDM model. As discussed in Paper I there is not yet a consensus on the interpretation of these types of measurements, with some authors reporting tension with ΛCDM. This situation is likely to change in the near future, e.g. via cross-correlations of the new ACT lensing maps and galaxy surveys and redshift-space distortion measurements with the dark energy spectroscopic instrument (DESI; DESI Collaboration 2016). Should such measurements conflict with ΛCDM on linear scales, our proposed solution of the S₈ tension will become untenable.

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

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

This paper is structured as follows. Section 2 discusses the DES, the data used, and the modelling choices made in our analysis. We compare the DES and KiDS constraints on the parameter A_mod in Section 3. Section 4 explores constraints imposed by the DES measurements on the redshift dependence of the power suppression. In Section 5, we explore the scale dependence of the suppression to provide more stringent tests of whether baryon feedback can resolve the S₈ tension. Finally, we summarize our conclusions and discuss the implications of our results in Section 6.

2 DARK ENERGY SURVEY

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

We reanalyse the DES cosmic shear data using the public DES pipeline and following the analysis of DES22.² We update the analysis choices as follows: we assume three neutrino species with two massless states and one massive state with a mass of 0.06 eV; we adopt the non-linear alignment (NLA) model for intrinsic alignments as described in Secco et al. (2022); we use hmcode-2020 (Mead et al. 2021) to model the non-linear matter power spectrum, which generates a non-negligible shift in the S₈ posterior compared to the halofit model (Takahashi et al. 2012) used in DES22. These analysis choices are tested and compared in detail in DES & KiDS Collaborations (2023).

Table 1 lists the priors used in this paper. For the ‘Free’ cases, we applied uninformative priors on the cosmological parameters, following DES22, with the exception of the neutrino mass which is kept fixed in our analysis. The remaining entries in Table 1 give the priors for the redshift calibration for each bin, Δz, the shear calibration for each bin, m, and the single intrinsic alignment parameter A₁ of the NLA model, as in DES22. We verify that leaving that neither freeing these nuisance parameters, nor fixing the alignment parameters to a different value significantly impacts the A_mod result. Furthermore, we test the more complex tidal alignment and tidal torquing (TATT) model for intrinsic alignments and find that the value of A_mod is not substantially changed, although the uncertainty on the parameter constraint increases.

In Paper I, which was exploratory in nature, we demonstrated that the best-fitting Planck ΛCDM cosmology provides acceptable fits to the KiDS shear–shear statistics if A_mod ∼ 0.7. However, we did not account for any uncertainty on the cosmological parameters, particularly S₈, as measured by Planck. As a consequence, Paper I overestimated the suppression of small-scale power required to resolve the S₈ tension with KiDS. In this paper, instead of a joint analysis, we include the Planck uncertainty on S₈ by applying a prior on S₈ and Ω_m derived from the base ΛCDM Planck TTTEEE chains,

This prior is described in more detail in Appendix  A. The scalar spectral index and physical baryon density are poorly determined by DES and KiDS but are determined to high-precision by Planck within the context of the six-parameter ΛCDM cosmology. Therefore, we fix these parameters to the best-fitting Planck values from Efstathiou & Gratton (2021) n_s = 0.9671 and ω_b = Ω_bh² = 0.02226. Finally, for a given value of Ω_m, the Hubble parameter h is set from the well-determined parameter combination Ω_mh³ = 0.09612. Applying a prior in this way allows us to accurately account for the Planck uncertainty without needing to evaluate the Planck likelihood in our analysis.

Results of analysing the DES Y3 ξ_± data for the different parameterizations and modelling choices discussed in this paper are summarized in Table 2. The first two entries (labelled 1 and 2), summarize results from DES22 (see their table 3) using their ΛCDM Optimized scale cuts (SCs), which excludes data points lying within the shaded regions in Fig. 1 from the analysis. Both of these analysis variants use halofit to model the non-linear power spectrum. Variant 1 uses the 5-parameter TATT model for intrinsic alignments and allows the neutrino mass to vary, as described in DES22. In variant 2, the neutrino mass is fixed to 0.06 eV and the two-parameter NLA model, a subspace of the TATT model which includes a redshift dependence, is used to model intrinsic alignments, resulting in a ∼1σ shift towards a higher value of S₈.

Variant 3 in Table 2 uses hmcode-2020 dark matter model for the non-linear power spectrum, instead of halofit and simplifies the intrinsic alignment model even further using the NLA model without a redshift-dependence [which Secco et al. (2022) have shown gives an acceptable fit to the DES Y3 data]. As in variant 2, the cosmological and nuisance parameters are allowed to vary over the prior ranges given in Table 1. With this set of analysis choices, the mean posterior value of S₈ in variant 3 is about 1σ higher than the value from variant 2. We refer the reader to DES & KiDS Collaborations (2023) for a detailed analysis of the impact of each these modelling strategies.

Variant 4 is the same as variant 3 except that it uses ξ_± measurements over the entire angular ranges plotted in Fig. 1.³ The best-fitting models from variants 3 and 4 are compared to the DES Y3 measurements in Fig. 1 and although baryon feedback is not accounted for in variant 4, they both provide excellent (almost indistinguishable) fits to the data. The χ² values for these fits are given in Table 2.

Comparing the posterior widths of S₈ with the Planck TTTEEE result of S₈ = 0.828 ± 0.016 (Efstathiou & Gratton 2021), we see that in variants 3 and 4, S₈ is lower by about 0.9 and 1.8σ, respectively. One might therefore conclude that with the choices made in this paper (i.e. adopting the NLA intrinsic alignment model and hmcode-2020) the DES Y3 weak lensing measurements are consistent with Planck ΛCDM even if baryonic feedback effects are ignored. This inference holds, apparently, even for variant 4, which has a smaller error on S₈ and is sensitive to spatial scales at which all cosmological hydrodynamical simulations show power spectrum suppression caused by baryonic feedback. However, focusing on the S8 parameter gives a misleading impression of consistency because the weak lensing constraints on other cosmological parameters are substantially weaker than Planck, it means that regions of cosmological parameter space allowed by DES can be disfavoured by Planck (see, for example appendix F of Amon et al. 2023). As we will show in the next section, suppression of the non-linear power spectrum is required at high significance to reconcile the Planck ΛCDM cosmology with DES (and KiDS) weak lensing measurements.

3 SMALL-SCALE POWER SUPPRESSION: CONSTRAINTS ON THE PARAMETER A_MOD

Following Paper I, in this section we incorporate the A_mod parameter to model suppression of power on non-linear scales via equation (1). Variants 5 and 6 in Table 2 use the full range of angular scales plotted in Fig. 1. Variant 5 allows the cosmological parameters to vary freely over their priors, i.e. it is identical to variant 4 but with the addition of the A_mod parameter. Variant 6 includes the Planck prior of equation (2).

Fig. 2 (left-hand side) compares the DES constraints from variants 5 (red) and 6 (yellow) in the A_mod − S₈ plane. As shown in Paper I and in Fig. 2, the A_mod parameter is strongly degenerate with the linear theory value of S₈. The vertical dashed line in Fig. 2 shows the best-fitting Planck ΛCDM value S₈ = 0.828. The inclusion of the Planck prior tightens the constraints significantly.⁴ One can see that adding the Planck prior shifts the best-fitting value of S₈ downwards, but within about 1σ of the best-fitting ΛCDM value measured by Planck alone. The Planck ΛCDM cosmology is therefore compatible with the DES Y3 weak lensing measurements provided that the power spectrum is suppressed on non-linear scales with

Thus, if no SCs are applied, A_mod differs from unity by about 2.7σ. Note, that the best-fitting model in variant 6 is plotted as the orange line in Fig. 1 and is almost indistinguishable from the best-fittings to variants 3 and 4 which allow cosmological parameters to vary freely.

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

The error bar increases compared to equation (3) and now the parameter A_mod differs from unity by only about 0.8σ. This, of course, does not conflict with equation (3) and tells us merely that large angular scales are not as sensitive to power spectrum suppression as smaller angular scales.

The choice of SCs in the DES Y3 analysis was based on the baryonic feedback effects measured from the EAGLE (McAlpine et al. 2016) and from the OWLS-AGN (van Daalen et al. 2011) simulations, as detailed in Krause et al. (2021) and Secco et al. (2022). Using these simulations, SCs were determined which resulted in a maximum two-dimensional bias in the Ω_m − S₈ plane of 0.14σ_2D for the cosmic shear analysis. DES22 reported cosmological constraints by applying these SCs instead of modelling baryonic feedback effects. Our results in equation (4) suggest that this strategy does largely reduce the sensitivity of cosmological results to power suppression on non-linear scales, since the value of S₈ in variant 3 is within 0.8σ of the Planck ΛCDM value. However, as SCs are used to mitigate baryonic effects, with no attempt to model feedback, the exact biases introduced into the S₈ parameter will depend on the accuracy of the OWLS-AGN simulation as an upper bound on baryonic feedback effects. One cannot rule out small biases towards low values of S₈ if the baryonic feedback is actually stronger than in these simulations (or if the power spectrum suppression is caused by the properties of the dark matter). The results in equation (4) perhaps hint that this might be the case. What is clear, however, is that when the SCs are removed, significant suppression of the non-linear spectrum is required to reconcile the DES Y3 lensing results with the Planck ΛCDM cosmology in our hypothesis. Recent work, using DES weak lensing measurements without SCs, by Arico et al. (2023) report constraints (see also Chen et al. 2022) using the baryonification prescription (Schneider & Teyssier 2015; Aricò et al. 2020) to model baryonic feedback but choose priors that restrict the strength of the feedback.

The values of A_mod in equations (3) and (4) are both higher than the value reported in Paper I from an analysis of KiDS weak lensing. We compare KiDS with the result of equation (3), where no SCs are used in either case (though we note that KiDS measurements extend to smaller angular scales than DES). In the analysis of KiDS reported in Paper I, to infer a value of A_mod we kept the ΛCDM cosmological parameters fixed to the Planck best-fitting values, whereas in this paper we have applied the Planck prior of equation (2). Including a Planck prior skews A_mod towards higher values since the joint likelihood peaks at slightly lower values of S₈. We have repeated the analysis of Paper I using ξ_± including the Planck prior (variant 13 in Table 2) finding

which is ∼1σ higher than the value reported in AE. The joint constraints on A_mod and S₈ from this variant are shown by the filled blue contours in Fig. 2. The constraints from KiDS where cosmological parameters are allowed to vary freely (variant 12) are shown by the blue dotted contours. These constraints are similar to those from DES, though displaced to lower values of S₈. The differences between KiDS and DES are consistent with sampling fluctuations. Naively combining the two estimates equations (3–5) we find

Thus, according to our model, to reconcile the Planck base ΛCDM cosmology with DES + KiDS weak lensing data requires power spectrum suppression on small scales at high statistical significance (∼4σ).

The power spectrum suppression corresponding to equations (3) and (5) at z = 0 is shown in the right-hand panel of Fig. 2. In this and subsequent figures, we compare our predictions to the power spectrum suppression measured from hydrodynamical simulations taken from van Daalen, McCarthy & Schaye (2020). The solid and dotted lines correspond to the lower and upper bounds on the power spectrum suppression suggested by the baryons and haloes of massive systems (BAHAMAS; McCarthy et al. 2017) simulations. These correspond to their AGN feedback parameter of log₁₀(ΔT_heat/K) = 7.6 and 8.0, respectively. With the feedback prescription adopted in BAHAMAS, the choice log₁₀(ΔT_heat/K) = 7.8 provides a good match to the observed gas fractions in groups and clusters and to the stellar mass function at z = 0.1 (van Daalen et al. 2020). The feedback prescription used in the two COSMO-overwhelmingly large simulations (C-OWLS; Le Brun et al. 2014) is the same as in BAHAMAS and uses higher values of log₁₀(ΔT_heat/K). As a result, the power spectrum suppression in C-OWLS is stronger and extends to lower wavenumbers than in the BAHAMAS simulations. As discussed in van Daalen et al. (2020), the stronger feedback in these simulations resulted in deficit of galaxies with stellar masses below 10¹¹h⁻¹M_⊙ compared to observations. For reference, the power spectrum suppression in the OWLS-AGN simulation used to select the DES angular SCs is similar to that in the fiducial BAHAMAS simulations with log₁₀(ΔT_heat/K) = 7.8. The suppression in the EAGLE simulations is much weaker than any of the simulations plotted in Fig. 2, see Chisari et al. (2019).

To reconcile Planck ΛCDM with weak lensing based on DES and KiDS, the power spectrum on non-linear scales must be more strongly suppressed than in the BAHAMAS simulation with log₁₀(ΔT_heat/K) = 8 (which over the wavenumber range |$k \lesssim 1 \, h\, {\rm Mpc}^{-1}$| is well-approximated by A_mod = 0.9). Stronger power spectrum suppression, closer to the two C-OWLS models plotted in Fig. 2 is required to match the results of equations (3) and (5).

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

4 REDSHIFT DEPENDENT POWER SUPPRESSION

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

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

where the functions f_k(z) are

The matter power spectrum suppression according to this model is

and uses six parameters {A, B, |$z_{c_1}$|⁠, α₁, |$z_{c_2}$|⁠, α₂} instead of the single parameter A_mod of equation (1). We adopt flat priors on these parameters as summarized in Table 3.

Fig. 4 compares the power spectrum suppression constraints from variant 9 (purple contours) with those from the one-parameter A_mod model of variant 6 (orange contours). We show the results at a redshift of z = 0.25, since at this redshift the results are insensitive to our choice of the fitting function and priors. The purple contours span a slightly broader range than the orange contours, since in the C_mod(z) model, one can trade-off increased suppression at z ∼ 0.3 with decreases in suppression at higher and lower redshifts and vice-versa. We also show the suppression measured in the cosmological hydrodynamics simulations at z = 0.25. These are almost identical to the simulation results at z = 0 shown in Fig. 2. Fig. 3 shows the redshift evolution of this parameter.

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

5 SCALE-DEPENDENT SUPPRESSION

In this section, we explore the scale-dependence of the non-linear matter power spectrum suppression by replacing the A_mod parameter with five parameters, A_im separated in wavenumber k as follows:

where k is in units of hMpc⁻¹ throughout this section.

To begin, we examine the contribution of each of the k-bins to the ξ_±(θ) prediction of the DES Y3 data,⁵ shown in Fig. 5. We make several observations: (i) as is well-known at any given angular scale θ the ξ_± statistics mix of information from a wide range of k-scales; (ii) ξ₋ is significantly more sensitive to non-linear information compared to ξ₊; (iii) the bin covering linear scales (k < 0.1, plotted in blue) makes little contribution to ξ₋ over the measured angular range, though it is important for the larger θ values of ξ₊; and (iv) the DES SCs effectively remove sensitivity to wavenumbers k > 1, thus small angular scales that are within the SCs must be included to constrain the parameters A₄ and A₅.

We adopt flat priors on each A_i parameter, as summarized in Table 4, with the lower limits describing much more extreme power suppression than seen in cosmological hydrodynamical simulations. The posterior mean value of S₈ and reduced-χ² for the A_i analysis are given in Table 2 for both the free cosmology and Planck prior cases (variants 10 and 11, respectively). The mean value of the posterior and the standard deviations of the A_i parameters are reported in Table 4.

Fig. 6 shows the predicted power suppression in each k-bin (separated by grey vertical lines), compared to the fiducial A_mod for the fits that include the Planck prior. The constraints from the binned model track the general shape and amplitude of the one-parameter A_mod model. The main new result from this analysis is that power suppression of ∼3 per cent − 10 per cent spanning mildly non-linear scales (bin 2, spanning wavenumbers in the range 0.1 < k < 0.5) is required to reconcile the Planck ΛCDM data with the DES weak lensing data. It is not possible to avoid suppression in bin 2 by increasing the suppression at smaller scales, mainly because ξ₋ is dominated by bin 2 over the angular range θ ∼ 40 − 100 arcmin (see the green curves in Fig. 5).

Fig. 6 also shows the power spectrum suppression measured in the BAHAMAS and C-OWLS simulations (as in Fig. 2). Evidently, if baryonic feedback is responsible for the apparent S₈ tension, the analysis of this section shows that the feedback must propagate to scales k ≲ 0.3. This requires stronger feedback than in the BAHAMAS simulation with log₁₀(ΔT_AGN/K) = 7.8 favoured by (McCarthy et al. 2017), in agreement with the conclusions of Section 3.

6 DISCUSSION AND CONCLUSION

The aim of this investigation has been to assess whether the S₈ tension can be resolved, that is Planck ΛCDM cosmology can be made consistent with weak lensing observations by modifying the matter power spectrum on non-linear scales. Following Paper I, we have investigated constraints on the power suppression parameter, A_mod of equation (1), using DES Y3 cosmic shear data. In this analysis, we include a Planck prior describing their constraints on key cosmological parameters and the associated uncertainties.

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

The constraints on A_mod depend on the angular SCs applied to the ξ_± measurements. If the DES ‘ΛCDM-Optimized’ angular SCs are imposed on ξ_±, the cosmological constraints from DES data are degraded and are statistically compatible with the Planck cosmology. For this case, A_mod is consistent with unity, though with a large error.

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

To investigate the wavenumber dependence, we solved for amplitude suppression factors A_i in five logarithmically spaced bins. The results show that consistency between DES and Planck ΛCDM requires suppression on scales |$k \lesssim 0.3\ h/\rm {Mpc}$|⁠. This result is in agreement with our results for A_mod and shows that the requirement of the data for power suppression on these scales is not an artefact of the simple A_mod parameterization.

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

To compare the results from the CMB with those from weak gravitational lensing, it is necessary to extract the linear amplitude of the matter fluctuations from statistics that are dominated by non-linear scales. This requires an accurate model of the dark matter power spectrum on non-linear scales, including modifications caused by baryonic feedback. Incorrect modelling of the non-linear spectrum can therefore lead to an apparent tension between weak lensing estimates of S₈ and those measured from the CMB.

This is illustrated by the remaining entries in Fig. 7 which summarize results from the KiDS and DES Y3 weak lensing. One can see that there are varying degrees of tension with the CMB. The two entries labelled ‘KiDS θ > 0.5 arcmin’ (from Paper I) and ‘DES θ > 2.5 arcmin’ (variant 4) use the full angular ranges of ξ_± reported by the two surveys (i.e. no SCs). For these entries, Fig. 7 shows results on S₈ allowing cosmological parameters to vary freely, using hmc ode-2020 to model the non-linear spectrum and ignoring baryonic feedback. The ‘tension’⁶ with the Planck TTTEEE + lensing entry is ∼3.7σ for KiDS and ∼2.1σ for DES. If we apply the DES ΛCDM -Optimized SCs (variant 3) the tension with the Planck S₈ drops to ∼1σ. This suggests that the SCs have largely eliminated sensitivity to non-linear power spectrum suppression, at the expense of increasing the error on S₈, leading to consistency with Planck. The fact that DES S₈ in this case is slightly low may be because the effects of non-linear power spectrum suppression have not been eliminated entirely.

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

If the weak-lensing S₈ tension is caused by suppression of power on non-linear scales, what would we expect for other cosmological measurements? Fig. 8 shows a sketch of the wavenumber and redshift ranges covered by various cosmological observations. As noted in Paper I, all measures of S₈ at low redshift from linear scales should agree with the Planck cosmology. Specifically, CMB lensing measurements combined with baryon acoustic oscillations, such as recent measurements from advanced ACT are consistent with the Planck cosmology (Madhavacheril et al. 2023). Cross-correlations of CMB lensing with galaxy surveys can provide an important test of this hypothesis. Forthcoming analyses using the latest CMB lensing maps from ACT (Qu et al. 2023) and the South Pole Telescope (Omori et al. 2023) should agree with the predictions of the Planck ΛCDM model.

As reviewed in Paper I, redshift-space distortions measured from linear scales from current galaxy redshift surveys are consistent with our interpretation, but not to high precision. This situation should change decisively in the near future with RSD measurements from DESI (see e.g. Schlegel et al. 2022, and references therein). Measurements of S₈ from quasar Ly-alpha lines from DESI uniquely give access to high k and z ≳ 1 and offer an opportunity to potentially break the degeneracy between baryonic feedback and non-standard dark matter. Tests of a redshift dependent growth, σ₈(z) or S₈(z) using measurements from from linear scales should be consistent with ΛCDM (García-García et al. 2021; DES Collaboration 2022; White et al. 2022).⁷

Weak galaxy lensing measurements offer a window to a wide range of scales as indicated in Fig. 8. In the future, it may be possible to reconstruct the power spectrum as a function of wavenumber and redshift from weak lensing data. Assuming that systematic errors can be sufficiently mitigated, it may become possible to accurately test the ΛCDM cosmology on linear scales and to reconstruct the matter power spectrum on small scales using weak galaxy lensing data alone. Such an analysis would establish unambiguously whether the S₈ tension is caused by a deviation from the ΛCDM model at late times, or whether it is an apparent tension caused by physics on non-linear scales.

ACKNOWLEDGEMENTS

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

This project has used public archival data from both the Dark Energy Survey and the Kilo Degree Survey. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Centre for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Centre for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy Survey. The collaborating institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, the Ohio State University, the OzDES Membership Consortium, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. Based in part on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017, 177.A-3018, and 179.A-2004, and on data products produced by the KiDS consortium. The KiDS production team acknowledges support from: Deutsche Forschungsgemeinschaft, ERC, NOVA, and NWO-M grants; Target; the University of Padova, and the University Federico II (Naples).

DATA AVAILABILITY

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

Footnotes

References

APPENDIX A: PLANCK ΛCDM PRIOR

In this appendix we justify the Planck prior of equation (2). In the base ΛCDM model, the spectral index n_s is well determined by Planck and can be fixed to the Planck best-fitting value. The main parameters that affect weak lensing are S₈, Ω_m, and H₀ (or combinations of these parameters). The parameter combination Ω_mh³ is a proxy for the acoustic peak location parameter, θ_*, and is extremely well determined by Planck. The Planck ΛCDM degeneracies in the space of S₈, Ω_m, and H₀ can be approximated accurately by adopting a two-dimensional Gaussian in S₈ and Ω_m and imposing a delta function constraint on the parameter combination Ω_mh³. The specific form of equation (2) comes from the inverse of the covariance matrix of the parameters S₈ and Ω_m determined from the cosmomc chains. The degeneracies in the S₈ − Ω_m and S₈ − H₀ planes inferred from equation (2) are plotted in Fig. A1 and are compared to samples from the cosmomc chains.