KiDS-Legacy: Cosmological constraints from cosmic shear with the complete Kilo-Degree Survey

Introduction

Our current understanding of the large-scale structure (LSS) of the Universe, obtained over the last three decades thanks to increasingly sophisticated astronomical observations, can be regarded as one of the great successes of modern scientific inquiry. Detailed observations of the LSS probe fundamental physics, since the structures we observe were formed by an interplay of physical effects. These primarily include the self-gravity of matter, cosmic expansion, astrophysical feedback processes, and the repulsive effect of the cosmological constant or dark energy.

When it comes to understanding the growth and evolution of LSS, the cosmological stage is set by observations of the cosmic microwave background (CMB) that reveal the structure of dark and baryonic matter at early times. Together with other probes, including big-bang nucleosynthesis (BBN) and baryon-acoustic oscillations (BAOs), the CMB has been used to establish and tightly constrain the so-called standard model of cosmology, $\Lambda$CDM. It is a strikingly simple model based on only a handful of numerical parameters ([103] (Qu et al. 2024; Ge et al. 2025)) and characterised, in particular, by two components that are at the origin of its name: a cosmological constant ($\Lambda$, which dominates the energy landscape at late times) and a cold dark matter (CDM) component that is dominant over baryonic matter by a factor of roughly five. Observations of the CMB constrain the parameters of this model at high redshift ($z \sim1100$), when the Universe was approximately 370 thousand years old. However, the model can be subsequently used to predict the properties of the LSS at late times ($z \lesssim2$; an extrapolation over more than 10 billion years), which can then be tested with low-redshift probes of cosmic structures.

Of the low-redshift probes, the relative prevalence of CDM poses a problem for those that rely on luminous tracers, such as galaxy clustering measurements. Modelling and constraining the relationship between luminous and dark matter (DM), also known as galaxy bias, constitutes one of the major systematic challenges of such measurements. In contrast, observations of the weak gravitational lensing effect of the LSS are intrinsically sensitive to all matter (dark or otherwise), eliminating the systematic uncertainty of galaxy bias. This unique feature has turned such ‘cosmic shear’ experiments into a major cosmological probe that is just entering its golden age. Indeed, it is the combination of cosmic shear with galaxy clustering – and their cross-correlation called galaxy-galaxy lensing (GGL) – that will potentially yield the most precise test of the standard model.

Over the past decade, the comparison between observations of low-$z$ LSS and their CMB-based predictions has revealed a tension in the $S_8 = \sigma_8\sqrt{\Omega_{\mathrm{m}}/0.3}$ parameter. Therefore, it has been the focus of extensive study. All contemporary cosmic shear measurements to date, except one, have yielded values of $S_8$ that are lower than CMB predictions, indicating a universe with less structure and/or matter. The significance of these measurements varies, but there is a clear tendency towards lower values in the cosmic shear results of the past decade. Only the Deep Lens Survey (DLS; Jee et al. [64]) has reported a value that is higher than that of Planck, and even then only slightly. In contrast, lower values have been reported by the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS; Heymans et al. [50]), as well as from previous results from the ongoing ‘stage-III’ surveys by the Kilo-Degree Survey (KiDS; [8] (Li et al. 2023a)), the Dark Energy Survey (DES; [5, 121]), the Hyper-Suprime Camera survey (HSC; [27] (Li et al. 2023c)), and a joint analysis by Dark Energy Survey and Kilo-Degree Survey Collaboration et al. (2023). Other low-$z$ LSS probes typically also share this trend but again with a varying level of significance (see Abdalla et al. [1], for an overview).

Cosmic shear probes relatively low redshifts, and relatively small scales compared to other probes of LSS (see e.g. Figure 1 of Broxterman & Kuijken [21], for a graphical summary). This makes cosmic shear somewhat more sensitive than other low-$z$ probes to non-linear structure formation and astrophysical effects: the latter mainly involve feedback processes driven by supernovae and active galactic nuclei, collectively referred to below as baryonic feedback. It has therefore been suggested that inaccuracies in the modelling of non-linear scales might be the reason for the tension in $S_8$, rather than a more fundamental cosmological cause ([143, 142, 4, 104]).

A definitive answer to the reality of the $S_8$ tension is within reach. Currently, three stage-III surveys have finished their observations and are preparing their final analyses; one of which we present here. The next generation of surveys, called stage-IV, are just beginning, represented by the ESA Euclid mission [33], the Legacy Survey of Space and Time (LSST) conducted with the Vera C. Rubin Observatory (Ivezić et al. [61]), the NASA Roman Space Telescope (Spergel et al. [126]), and the Chinese Space Station Optical Survey (Gong et al. [45]). We note that Euclid is expected to yield first cosmologically relevant results in 2026. These stage-IV surveys will soon provide datasets capable of measuring the growth of structure at low $z$ with unprecedented precision.

New facilities and new instruments invariably bring along new challenges owing to, for instance, the novelty of their hardware. Comparatively, the final (usually called ‘legacy’) releases of a survey typically represent the culmination of years or decades of analysis, understanding, and refinement regarding their survey and the associated data. As such, the final releases of stage-III surveys will provide a crucial insight and reference for subsequent lensing studies of LSS with stage-IV surveys.

Here, we present the cosmic shear analysis of the completed Kilo-Degree Survey (KiDS) based on the fifth and final data release (KiDS-DR5; Wright et al. [140]), dubbed KiDS-Legacy. With extensive wavelength coverage enabled by its unique pairing with its near-infrared (NIR) sister survey VIKING (VISTA Kilo-Degree Infrared Galaxy Survey) and with its superb image quality in the $r$-band, KiDS is the stage-III survey that is arguably most suited to simultaneously leverage high-redshift galaxies and accurately calibrate their redshift distributions. Over the past decade, KiDS has been pushing to increasingly high redshifts, while continuously improving the redshift calibration, shear measurements, shear calibration through image simulations, and methodology to accurately extract cosmological constraints from the imaging data.

This paper represents the culmination of these efforts, providing the most robust and precise analysis of KiDS to date. Our dataset is the largest analysed by KiDS thus far (Sect. 3.1), utilising the most extensive spectroscopic redshift compilation prepared for any cosmic shear analysis (Sect. 3.2), and has been calibrated using the most sophisticated simulations produced for this purpose (Sect. 3.4). Improvements in methodology (Sect. 2) have allowed us to access the highest redshifts (with the most tomographic bins) studied for cosmic shear to date (Sect. 3.4). An efficient workflow for this study was enabled by the new CosmoPipe analysis framework, which we have also made publicly available. We present null tests of our dataset in Sect. 4. The cosmological results are presented in Sect. 5, discussed in Sect. 6, and summarised in Sect. 7. Additionally, we include supplementary information regarding our power spectrum emulation (Appendix A), on the definition of parameters for our fiducial intrinsic alignment model (Appendix B), construction of band power spectra (Appendix C), covariance validation (Appendix D), $B$-mode exploration (Appendix E), correlation function cosmological analyses (Appendix F), detailed posterior analyses (Appendix G), tables of cosmological constraints (Appendix H), and details of changes made to the analysis and manuscript post-unblinding (Appendix I). Finally, this paper has been released alongside two companion papers: the work of Stölzner et al. [128] presents an extensive analysis of internal and external consistency tests of the KiDS-Legacy dataset (including joint constraints with other datasets) and the work of Wright et al. [138] describes the various redshift distribution and bias estimation developments made for KiDS-Legacy.

Modelling and theoretical framework

The main cosmic shear observable is the ellipticity correlation between pairs of galaxies. The signal depends on the two-point statistics of the shear and galaxy intrinsic alignment fields, which both trace the matter distribution and therefore depend on cosmological information. For our predictions and sampling, we used CosmoSIS, a modular cosmology pipeline described in Zuntz et al. [144]. We included a number of new and/or modified modules in our analysis, including a CosmoPower emulator for non-linear power spectra (Appendix A) and new intrinsic alignment models (Sect. 2.4). In this section, we start with the derivation of cosmic shear power spectra from matter power spectra and then detail our choices of intrinsic alignment models. Next, we introduce our observable two-point statistics, briefly discussing their measurement and prediction methodology. We end this section with a brief review of our analytical covariance modelling.

Shear power spectrum

Our two-point statistic predictions are calculated from shear (cross-)power spectra of galaxy samples $i$ and $j$, $C_{\mathrm{GG}}^{(ij)}(\ell)$, where G stands for ‘gravitational’, which distinguishes these power spectra from intrinsic alignment contributions to the two-point signal. Shear power spectra are given by Limber-approximated projections (Kaiser [70]; LoVerde & Afshordi [89]; Kilbinger et al. [72]) taking the form

where $P_{\mathrm{m,nl}}$ is the non-linear matter power spectrum and $f_K(\chi)$ is the comoving angular diameter distance. Although the integral in Eq. (1) technically runs over the entire line of sight to the horizon, $\chi_{\mathrm{hor}}$, in practice, the weak lensing kernel, $W_{\mathrm{G}}^{(i)}(\chi)$, limits the integral to the comoving distance of the furthest galaxy in our sample, expressed as

where $n_{\mathrm{S}}^{(i)}(\chi)$ is the distribution of comoving distances of galaxies in source sample $i$; in practice, we express this in terms of redshift, namely, $N(z)\,\mathrm{d}z=n_{\mathrm{S}}(\chi)\,\mathrm{d}\chi$. Here, $\Omega_{\mathrm{m}}$ is the total matter density parameter today, $H_{0}$ is the Hubble constant, $c$ is the speed of light, and $a(\chi)$ is the scale factor normalised to unity today.

Matter power spectrum

In our cosmic shear analysis, we remained mostly sensitive to Fourier scales $0.1\,h/\mathrm{Mpc}\lesssim k\lesssim1\,h/\mathrm{Mpc}$, with a weighting of scales that depends on the two-point statistic (see Sect. 2.5). These scales are well within the (quasi-) non-linear regime and therefore require an accurate non-linear power spectrum model for their predictions. Currently, there are multiple approaches that can produce non-linear matter power spectra, with varying levels of accuracy, for this range of scales. Here, we followed the most popular approach for cosmic shear analyses, using a method inspired by the halo model and calibrated against simulations. Specifically, we used the Boltzmann solver CAMB (Lewis et al. [85]; Howlett et al. [60]) to estimate the linear matter power spectrum and the augmented halo model ‘HM-Code2020’ (Mead et al. [95]) to derive its non-linear evolution. HMCode2020 has a similar level of accuracy as popular emulators trained directly on N-body simulations (2.5%; see e.g. Knabenhans et al. [73]; Aricò et al. [6]) and is twice as accurate as HALOFIT from Takahashi et al. [129], which has been used in a number of previous cosmic shear analyses (see e.g. Amon et al. [5]; Hikage et al. [53]). Dark Energy Survey and Kilo-Degree Survey Collaboration et al. (2023) presented a comparison of HMCode2020 and Halofit with EuclidEmulatorv2, showing that HMCode2020 produces unbiased results when analysing mock data created with EuclidEmulatorv2; meanwhile Halofit tends to overestimate power for lower values of $S_8$, resulting in an underestimated $S_8$ value. Similarly, an earlier version of HMCode, HMCode2015 [95], which was used in previous KiDS analyses, is $\sim30\%$ less accurate than HMCode2020. This can subsequently produce small differences between cosmological constraints estimated using the two codes: Tröster et al. [132] demonstrated that the most important difference between these versions of HMCode is in their treatment of baryon feedback, which results in a slightly higher $S_8$ using HMCode2020. Lastly, we note that the parameter values at which HMCode was fitted to simulations are somewhat more constrained than our chosen priors for $h^2\Omega_{\mathrm{m}}$ (see Mead et al. 2021, and Table 3). However, given that the fitting functions of HMCode are based on physically motivated parameters, we did not anticipate any significant issues just beyond this range. Furthermore, we note that our analysis remained unaffected by this choice.

Baryonic feedback

The primary features of the matter power spectrum can be determined by assuming all matter is in the form of (pressureless) CDM, with modifications due to processes involving baryonic matter. At large scales ($0.05\ h/\mathrm{Mpc} \lesssim k \lesssim0.5\ h/\mathrm{Mpc}$), BAOs produce oscillations in the power spectrum. Over a wide range of intermediate scales ($0.1\ h/\mathrm{Mpc} \lesssim k \lesssim10\ h/\mathrm{Mpc}$), power is suppressed through feedback processes from active galactic nuclei (AGN), reaching a maximum suppression at around $k \sim5\ h/\mathrm{Mpc}$. At even smaller scales ($k \gtrsim10\ h/\mathrm{Mpc}$), star formation processes lead to a sharp increase in power (see e.g. Semboloni et al. 2011).

To model baryonic feedback processes, HMCode2020 performs a modification to the concentration-mass relation of the halo profiles as a function of their gas fraction; this, in turn, depends on the strength of the AGN feedback (parametrised by $\log(T_{\mathrm{AGN}}) \in[7.1, 8.3]$). This feedback model is then calibrated against a suite of hydrodynamical simulations from BA-HAMAS [93], which had been generated using a wide range of effective feedback amplitudes and cosmological models. Recently, Schaye et al. (2023, FLAMINGO) presented a new suite of hydrodynamical simulations, including emulators for generating power spectra [112], as a mechanism for modelling baryonic effects. Moving beyond hydrodynamical simulations, Salcido et al. (2023, SP(k)) presented alternative phenomenological models for the same purpose, while Schneider & Teyssier [115]; Aricò et al. [7] presented perturbative models that act directly on the outputs of dark-matter only simulations and can be empirically calibrated [46, 114]. We did not utilise these models for our cosmological analyses here, as these models were published over the course of the analysis of KiDS-Legacyor were not (at time of writing) implemented within our Cosmosis sampling framework. However, we note that a reanalysis of our data with these models is a natural and straightforward project that will be undertaken in a future publication.

Intrinsic alignments

The intrinsic alignment (IA) of galaxies is correlated with the observed ellipticities of galaxy images, making them indistinguishable from the cosmic shear effect we wish to probe for cosmological parameter estimation. In the limit of small shears and IA, the two-point statistics of galaxy ellipticities are given by

where $C_{\mathrm{GG}}^{(ij)}$ is the cosmic shear signal given by Eq. (1). Analogous Limber equations can be derived for IA contributions, which are typically grouped into two categories. First, intrinsic-intrinsic (II) correlations represent the correlated alignment of galaxies that are tidally aligned by the same ambient LSS:

Second, shear-intrinsic (GI) correlations are driven by the mutual gravitational effects of LSS on (i) galaxies within the structures, and (ii) the lensed images of background sources that these structures produce [57]:

with $C_{\mathrm{IG}}^{(ij)}=C_{\mathrm{GI}}^{(ji)}$. In Eqs. (4) and (5) the power spectra $P_{\mathrm{II},\mathrm{X}}$ and $P_{\mathrm{\delta I},\mathrm{X}}$ are given by the chosen IA model. We note that the IA contributions feature a more compact kernel along the line of sight than the lensing kernel in Eq. (1), and hence the Limber approximation is poorer for these signals. However, this effect is not significant for our analysis, as the overall IA contributions to the total signal is small, making up only around 10 percent of the signal in our analysis. Additionally, we are only probing scales $\ell\gtrsim50$, where the Limber approximation is sufficient even for more compact kernels [84].

Since IA are caused by the local matter environment, they have a different redshift scaling than lensing, which however is insufficient to isolate the cosmological signal without substantial information loss [67]. Therefore, cosmic shear analyses opt for physically motivated IA models that are inferred jointly with the cosmology (see Lamman et al. 2024, for a review of the field). Guided by this, we implement four IA models in KiDS-Legacy: a widely used baseline with a single overall amplitude parameter (NLA, Sect. 2.4.1), two minimal extensions of NLA with either an additional redshift (NLA-z, Sect. 2.4.2) or scale dependence (via our restricted TATT model, NLA-k; Sect. 2.4.3), and a version of NLA that explicitly incorporates the well-established trends of IA with galaxy type and host halo mass (NLA-M, Sect. 2.4.4). The latter model is the fiducial model chosen for our cosmological analyses.

The non-linear linear alignment (NLA) model

The so-called NLA model [19] assumes that the intrinsic ellipticity of a galaxy depends linearly on its surrounding tidal quadrupole [24]. Moreover, it is commonly assumed that it is the tidal field at some early time during the formation of the galaxy that sets the amplitude of alignment [57]. This leads to the following expressions for the IA power spectra,