Going beyond $S_8$: fast inference of the matter power spectrum from weak-lensing surveys

Abstract

Weak lensing surveys are often summarized by constraints on the derived parameter ${S_8\equivσ_8\sqrt{Ω_{\rm m}/0.3}}$, obscuring the rich scale and redshift information encoded in the data, and limiting our ability to identify the origin of any tensions with $Λ$CDM predictions from the cosmic microwave background. In this work, we introduce a fast and flexible framework to extract the scale-dependent matter power spectrum $P(k, z)$ from cosmic shear and CMB lensing measurements, parameterizing deviations from the Planck $Λ$CDM prediction as a free function $α(k)$. Using public data from DES Y3, KiDS-1000, HSC Y3, and ACT DR6, we constrain $α(k)$ with fast Hamiltonian Monte Carlo inference, employing multipoles up to $\ell_{\rm max}\sim2000$ for the galaxy lensing surveys. Our results show a consistent 15-30% suppression in the matter power spectrum at intermediate scales ($k \sim 0.1-1{\rm Mpc}^{-1}$) in galaxy-lensing data relative to a Planck $Λ$CDM prediction with a CDM-only (no baryonic feedback) power spectrum, with combined tensions reaching up to $4σ$. This is under a fixed cosmology and with analytic marginalization over shear and redshift calibration uncertainties. In contrast, ACT CMB lensing is consistent with $Λ$CDM at ${k\lesssim 0.1 {\rm Mpc}^{-1}}$. We validate our method using mock data, quantify consistency between datasets, and demonstrate how the resulting $α(k)$ likelihoods can be used to test specific models for the power spectrum. All code, data products, and derived likelihoods are publicly released. Our results highlight the importance of reporting lensing constraints on $P(k, z)$ and pave the way for model-agnostic test of growth of structure with upcoming surveys such as LSST, Euclid, and Roman.

Figures (15)

• Figure 1: The contours in the main panel show the signal-to-noise ratio (SNR) defined in \ref{['eq:snr']} and hence the sensitivity of each experiment on the scale-redshift $(k,z)$ plane. The upper and right panels show the SNR projected along each direction, with dashed lines denoting the mean redshifts and (log) scales that the experiments are sensitive to (mean values are provided in \ref{['tab:kzeff']}). Where the contours overlap, they probe the same physics in redshift-independent $\alpha(k)$. However, as CMB- and galaxy-lensing surveys have little overlap, there are conceivable models that may impact the $P(k,z)$ probed by one observable but not the other. Therefore, we urge caution when combining these data sets on $P(k,z)$ and $\alpha(k)$. The HSC contours exclude scales following the original HSC analysis, such that all galaxy surveys have $\ell_{\rm max} \simeq 2000$. When including all HSC scales, the sensitivity extends to smaller scales, as expected. We also account for the ACT fiducial scale cuts, with $\ell_{\rm max}=763$. Note that the contours are simply linearly spaced from the maximum SNR to $0$, these do not represent $\sigma$ levels.
• Figure 2: Ratios of the posterior and prior widths for $\alpha(k)$ for DES using $n_k=24$$k$-bins, without (blue triangles) and with (red points) a smoothing prior. From the case without smoothing, we can determine the $k$-range that is constrained $\sim 3e-2\iMpc<k<1\iMpc$ shown by the unshaded band between the gray areas. We use the prior and posterior covariances to compute the effective number of constrained parameters, of order $n_{\rm eff}\simeq6.5$ for DES, on par with the 9 bins in the constrained range. Smoothing reduces the relative impact of data and the effective number of constrained bins to $n_{\rm eff}\simeq4.0$, but yields more physical curves.
• Figure 3: Correlation matrices of the posterior distribution of $\bm{\alpha}$'s for DES without smoothing (left), of the smoothing prior with smoothing strength $\sigma_{\rm s}=0.3$ (center), and of the posterior distribution for DES when including this smoothing prior (right). The middle and right panels demonstrate the strong correlations introduced across bins where data loses sensitivity at small and large $k$, in the regions beyond where $\alpha(k)$ are well-constrained.
• Figure 4: Validation tests performed using mock data generated at Planck$\Lambda$CDM cosmological parameters, incorporating various deviations to the standard nonlinear power spectrum. For each panel, the posterior of each component of $\bm{\alpha}=\qty(\alpha(k_1),\dots,\alpha(k_{n_k}))$ is visualized as a box. The vertical extent of the boxes represents the 68% limits of the posterior, computed with GetDist2019arXiv191013970L. The horizontal extent corresponds to the width of the logarithmic $k$-bins. Dark blue boxes represent the posterior with smoothing ($\sigma_{\rm s}=0.3$), while light blue boxes show the posterior without smoothing ($\sigma_{\rm s}=\infty$). The red lines indicate the relative differences between the injected and fiducial power spectra, evaluated at $z=0$ and at the effective DES redshift $z_{\rm eff}=0.36$. Note that any divergence between the posterior and smoothed posterior boxes in regions of low data sensitivity does not necessarily invalidate the recovery of the input cosmology. In the gray regions, the posterior distribution is dominated by the prior, and it is either uniform (no smoothing) or favors flat curves that satisfy the smoothing prior. Top panel: a deviation following the $A_{\rm mod}$ parametrization of 2206.11794 is successfully recovered. Middle panel: baryonic feedback was modelled using HMCode2021MNRAS.502.1401M and the injected deviations were recovered within the constrained $k$-range, denoted by the unshaded band between gray regions. Bottom panel: a $(\Sigma,\mu)$ modified-gravity model was implemented. While the deviation at the effective DES redshift ($z_{\rm eff}=0.36$) was recovered, the redshift-dependent deviation from the $\Lambda$CDM power spectrum was not captured by our only-scale-dependent model, as expected.
• Figure 5: Constraints from individual lensing surveys on $\alpha(k)$ defined as $P(k,z) \equiv (1+\alpha(k)) P_{\rm fid}(k,z)$ where $P_{\rm fid}(k,z)$ is the nonlinear $\Lambda$CDM matter power spectrum at Planck cosmology (see \ref{['tab:fiducial_cosmology']}) computed with halofit. \ref{['tab:kzeff']} shows the effective scales and redshifts probed by these experiments. Boxes are used to represent the 68% credible intervals derived from the posterior distributions of $\alpha(k)$ computed in 24 logarithmically-spaced $k$-bins. Lighter shaded boxes represent the unsmoothed, more strongly anti-correlated $k$-bins, while darker boxes show the smoothed, decorrelated bins (smoothing scale $\sigma_{\rm s}=0.3$). The gray shaded regions indicate where each dataset loses sensitivity, and where the points become strongly correlated (see \ref{['fig:DES_constraints', 'fig:DES_corr']}). We note that our model does not include intrinsic alignments and baryonic feedback, which could contribute to the lower power spectrum observed by cosmic shear surveys. Results adopting a DES Y3 cosmology 2203.07128 as fiducial are shown in \ref{['app:des_cosmo']}.