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Open Access
Issue
A&A
Volume 709, May 2026
Article Number A82
Number of page(s) 14
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361/202558581
Published online 05 May 2026

© The Authors 2026

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

The model currently best capturing the dynamics of the Universe on large scales is the Λ-cold dark matter model with vanishing spatial curvature (flat ΛCDM). Despite some discrepancies, this simple model with only a few parameters successfully explains a wide range of observations.

The weak gravitational lensing effect of the large-scale structure (LSS), also known as cosmic shear, has become a staple cosmological observable since its first detection over two decades ago (Kaiser et al. 2000; Wittman et al. 2000; Van Waerbeke et al. 2000; Bacon et al. 2000). Due to its sensitivity to the growth of structure as well as the background cosmology, it has the potential to probe the ΛCDM model and its extensions. The most recent analysis comes from the so-called Stage-III weak lensing surveys, the Kilo-Degree Survey (KiDS, see Wright et al. 2025b; Stölzner et al. 2025, for the most recent analysis)1, the Dark Energy Survey (DES, see e.g. Amon et al. 2022; Secco et al. 2022; Samuroff et al. 2023)2, and the Subaru Hyper Suprime-Cam Subaru Strategic Project (HSC, see e.g. Li et al. 2023c; Dalal et al. 20233). Cosmic shear is joined by other probes such as the cosmic microwave background (CMB; Planck Collaboration VI 2020; Camphuis et al. 2025; Louis et al. 2025), baryon acoustic oscillations (BAOs; Adame et al. 2025; Desi & Abdul-Karim 2025; Alam et al. 2021), redshift space distortion (RSD; Hou et al. 2021), expansion history measurements via Type Ia supernovae (SNeIa; Scolnic et al. 2022; Brout et al. 2022), as well as the present-day expansion rate, that is the Hubble constant, through Cepheids (Riess et al. 2019), strongly lensed quasars (Wong et al. 2020), or the tip of the red giant branch and J-region asymptotic giant branch stars (see Freedman et al. 2025, for an overview). These last measurements technically do not depend on cosmology, as they are local and thus cannot distinguish between different cosmological models beyond the overall normalisation of the expansion rate.

The first direct evidence of the accelerated expansion of the Universe (Riess et al. 1998; Schmidt et al. 1998; Perlmutter et al. 1999) established cosmological probes across cosmic time as a test bed for physics beyond the standard model. These measurements implied, in hindsight, an unsurprisingly non-vanishing second coupling constant, Λ, in the gravity sector. One can express Λ itself as an ideal fluid with constant energy density and with negative pressure. More generally, one can relate the energy density to the pressure via an equation of state, w[a(z)], which can depend on redshift z. This dynamic scenario can itself originate from an additional field that acts solely on the background cosmology and does not form perturbations. A key target of cosmology is therefore to use cosmological observables to infer w(a) and test whether it assumes its ΛCDM value of w = −1. Many constraints on w use an expansion up to linear order, which effectively approximates many different dark energy models (Chevallier & Polarski 2001; Linder 2003). Recently, the Dark Energy Spectroscopic Instrument (DESI Adame et al. 2025; Desi & Abdul-Karim 2025) found tentative evidence that w ≠ −1 in this parametrisation using BAO measurements and external probes including the CMB. This modification of the expansion history becomes particularly interesting in light of the Hubble tension, the discrepancy between local and early time measurements of the expansion rate (see e.g. Schöneberg et al. 2022; Kamionkowski & Riess 2023; Di Valentino et al. 2025, for reviews and summaries).

While Wright et al. (2025b) and Stölzner et al. (2025) recently put forward constraints on flat ΛCDM using the final data release of KiDS (DR5; Wright et al. 2024), the goal of this paper is to expand this analysis to minimal extensions to the baseline cosmological model and include external datasets. Unlike in the previous KiDS analyses (KiDS-1000 and its reanalyses; Asgari et al. 2021; Tröster et al. 2021; van den Busch et al. 2022; Li et al. 2023a; Yoon et al. 2025), which found a tension in the amount of matter clustering in the Universe compared to the CMB (the S8 tension), KiDS-Legacy is consistent with the other cosmological observables. The driving contributing factors are updated redshift calibration and scale cuts (see Appendix I in Wright et al. 2025b). This consistency allows the combination of different probes and opens the possibility of constraining extended parameter spaces.

On the data side, we used CMB primary anisotropies from Planck (Planck Collaboration V 2020), the Atacama Cosmology Telescope (ACT; Louis et al. 2025), and the South Pole Telescope (SPT; Camphuis et al. 2025), the signal from gravitational lensing of the CMB (Planck Collaboration V 2020; Madhavacheril et al. 2024; Ge et al. 2025), BAO data from DESI (Desi & Abdul-Karim 2025), RSD from the Extended Baryon Oscillation Spectroscopic Survey (eBOSS, e.g. Hou et al. 2021), SNeIa from the Pantheon+ sample (Scolnic et al. 2022), and combined cosmic shear measurements from KiDS-Legacy and DES Y3 based on Dark Energy Survey and Kilo-Degree Survey Collaboration (2023) with the updated data from Wright et al. (2024). For the modelling, we focused on three main extensions to the fiducial analysis: (i) varying neutrino mass (νΛCDM), (ii) dynamic dark energy (w0waCDM), and (iii) a spatially non-flat Universe (ΩKCDM). Although the linear growth factor is modified in dynamic dark energy scenarios, the extensions studied here do not explicitly alter the gravity sector; that is, we assume general relativity throughout this paper and introduce no new gravitational fields. In a companion paper (Stölzner et al. 2026), we investigate, in more detail, extensions that alter the growth of structures by modifying the gravitational law in the context of Horndeski theory.

We structure this paper as follows: Sect. 2 briefly summarises the methodology, the inference pipeline, data, and likelihood used in this analysis. In Sect. 3, we present the results of the three types of extended models for different probe combinations. Lastly, we summarise our findings and conclude in Sect. 4.

2. Methodology and datasets

In this section, we first briefly summarise the methodology used for the cosmic shear analysis, which follows the one used in Wright et al. (2025b), Stölzner et al. (2025), Reischke et al. (2025).

2.1. Cosmic shear and KiDS-Legacy

The KiDS-Legacy dataset combines two European Southern Observatory Public Surveys: KiDS (de Jong et al. 2013) on the VLT Survey Telescope (VST) and on the VISTA telescope, the VISTA Kilo-Degree Infrared Galaxy Survey (VIKING; Edge et al. 2013). The KiDS data span 1, 347 deg2 in two stripes on and south of the celestial equator, with optical bands u, g, r, i from VST and Z, Y, J, H, Ks from VIKING. Compared with the previous release, DR5 adds a second i-band pass and increases the area by 34 per cent. The KiDS-Legacy lensing catalogue contains 41 million source galaxies over 967 deg2, with an effective number density of 8.79 arcmin−2. Deeper i-band imaging and expanded spectroscopic calibration allow a higher photometric redshift limit of zB ≤ 2 and an extra high redshift tomographic bin. Sources are split into six roughly equally-populated bins by photometric redshifts, and the corresponding redshift distributions are calibrated against deep spectroscopy using self-organising maps and the multicolour SKiLLS simulations (see Li et al. 2023b; Wright et al. 2025a, for details). Galaxy shapes were measured with an updated version of the LENSFIT algorithm (Miller et al. 2013; Fenech Conti et al. 2017) and calibrated with SKiLLS.

We use CosmoPipe4 for the cosmic shear measurement, calibration, and inference infrastructure. As the summary statistic, we use the first six modes of the complete orthogonal sets of E/B-integrals (COSEBIs; Schneider et al. 2010; Asgari et al. 2012), measured from 2 to 300 arcmin. The COSEBI E-modes are integrals over the angular power spectrum of the observed ellipticities, C ϵ ϵ ( i j ) ( ℓ ) Mathematical equation: $ C^{(ij)}_{\epsilon\epsilon}(\ell) $ in tomographic bins i and j, which in turn can be written as

C ϵ ϵ ( i j ) ( ℓ ) = ∫ 0 χ H d χ W ϵ ( i ) ( χ ) W ϵ ( j ) ( χ ) f K 2 ( χ ) P m , nl ( ℓ + 1 / 2 f K ( χ ) , z ( χ ) ) . Mathematical equation: $$ \begin{aligned} C^{(ij)}_{\epsilon \epsilon }(\ell ) = \int _0^{\chi _{\rm H}} \mathrm{d}\chi \, \frac{W^{(i)}_{{\epsilon }}(\chi )W^{(j)}_{\epsilon }(\chi )}{f_{ K}^2(\chi )}P_{\rm m, nl}\left(\frac{\ell +1/2}{f_{ K}(\chi )},z(\chi )\right). \end{aligned} $$(1)

Here, Pm, nl is the non-linear matter power spectrum, which we model using HMCODE2020 (Mead et al. 2021). This employs a single-parameter (log10TAGN) baryonic feedback model to include the uncertainty of the modelling of baryonic feedback on scales k > 0.1h/Mpc. Furthermore, fK, χ, and χH are the comoving angular diameter distance, the comoving radial distance, and the comoving horizon distance, respectively. Lastly, the lensing kernel W ϵ ( i ) Mathematical equation: $ W^{(i)}_{\epsilon} $ contains the distribution of galaxies and the lensing geometry (see e.g. Wright et al. 2025b).

We adopt the fiducial mass-dependent intrinsic alignment (IA) model NLA-M (Wright et al. 2025b), an extension of the non-linear alignment (NLA) model that includes alignments for red, early-type galaxies and assumes zero alignment for blue, late-type galaxies. Wright et al. (2025b) investigated different alignment models and their impact on the inferred value for S8, finding changes of not larger than 0.2σ. Our choice is further supported by a red-blue split analysis conducted by Stölzner et al. (2025), which found vanishing alignment for blue galaxies. Early types are identified by spectral type5TB < 1.9, fitting six SED templates with interpolation; the cut selects galaxies with an elliptical spectral contribution. The IA alignment strength for red galaxies is modelled as a power law in the mean halo mass for each tomographic bin. We use two alignment parameters, AIA and β, which represent the amplitude of IAs and the slope of the IA mass scaling, respectively. In our analysis, we incorporate the joint posterior on AIA and β from Fortuna et al. (2025) as a prior, approximating it with a bivariate Gaussian distribution. Furthermore, we apply a multivariate Gaussian prior on the halo mass for each tomographic bin.

The cosmic shear measurements are combined with the DES Y3 data (Amon et al. 2022; Secco et al. 2022). As in Dark Energy Survey and Kilo-Degree Survey Collaboration (2023), roughly eight per cent of the DES data are removed to avoid overlap with KiDS. We do not use HSC Y3 data (Dalal et al. 2023; Li et al. 2023c); while the likelihood is publicly available, there is significant overlap between the surveys. Since the overall constraining power of HSC Y3 is weaker than the combined power of KiDS and DES, we do not expect this omission to change the results.

2.2. BAO and RSD data

We use BAO measurements (BAO scale, not including the Alcock-Paczyński effect) from DESI DR2 (Desi & Abdul-Karim 2025). DESI contains a bright galaxy survey (BGS, Hahn et al. 2023), luminous red galaxies (LRGs, Zhou et al. 2023), emission line galaxies (ELGs, Raichoor et al. 2023), the Lyman-α forest (du Mas des Bourboux et al. 2020), and quasars (QSOs, Chaussidon et al. 2023); cosmological BAO results are given in Adame et al. (2025). We employ the public DESI likelihood measurements for BGS (z ∈ [0.1, 0.4)), two LRG bins (z ∈ [0.4, 0.6) and z ∈ [0.6, 0.8), respectively), ELG (z ∈ [1.1, 1.6)), combined LRG+ELG (z ∈ [0.8, 1.1), QSO (z ∈ [0.8, 2.1)) and Lyman-α forest (z ∈ [1.77, 4.16)) samples. From the RSD measurements of the growth rate, we use the following data: the Sloan Digital Sky Survey DR7 main galaxy sample (Ross et al. 2015; Howlett et al. 2015), BOSS DR12 (Alam et al. 2021), eBOSS DR16 ELGs (Tamone et al. 2020; Raichoor et al. 2023; de Mattia et al. 2021), eBOSS DR16 LRGs (Bautista et al. 2021; Gil-Marín et al. 2020), eBOSS DR16 QSOs (Neveux et al. 2020; Hou et al. 2021).

2.3. CMB data

We use CMB data from Planck Collaboration V (2020), employing the compressed Planck likelihood of Prince & Dunkley (2019) where the ℓ < 30 temperature likelihood is approximated by two Gaussian points, and PLIK-LITE TTTEEE is used for ℓ > 30. We impose a Gaussian prior on the reionisation optical depth τ, derived from Planck base ΛCDM constraints, to mimic the constraints from low-multipole polarisation measurements. We note that the low-multipole polarisation measurements are not directly sensitive to the extensions studied in this paper. However, as discussed, for example, in Sailer et al. (2026), a larger optical depth would reduce some of the discrepancy with low-redshift probes. The origin of this remains unclear. For the CMB lensing signal, we employ the Planck lensing signal from Carron et al. (2022). We do not vary the nuisance parameter Alens in the fiducial analysis. This is motivated by the addition of CMB data where Alens was found to be consistent with unity (Camphuis et al. 2025). For an extended discussion of the effect of Alens, we refer to our companion paper (Stölzner et al. 2026).

Furthermore, we add ACT DR6 lensing and primary CMB (Qu et al. 2024; Madhavacheril et al. 2024; Naess et al. 2025; Louis et al. 2025; Calabrese et al. 2025) combined with SPT-3G D1 temperature and polarisation anisotropies (Camphuis et al. 2025). The SPT lensing signal is taken from SPT-3G 2018 (Pan et al. 2023). For these datasets, we use the CANDL likelihood (Balkenhol et al. 2024). To remove correlations between ACT and Planck, the Planck data are used only for ℓ < 1000 and ℓ < 600 in TT and TEEE, respectively. The correlations between Planck + ACT and SPT are negligible, as SPT-3G D1 only covers a small portion of the sky. Although some cross-correlation is present in the CMB lensing signals, Qu et al. (2026) showed that it has a negligible impact on the reported error bars. We therefore append the ACT and SPT data to our data vector at multipoles larger than the limits above.

Lastly, we neglect any cross-correlation between the cosmic shear and CMB lensing signals in the covariance matrix. Robertson et al. (2021), Yao et al. (2023) measured the cross-correlation of KiDS-1000 with the CMB lensing signal of the Atacama Cosmology Telescope, and Omori et al. (2023) did the same for DES Y3, Planck, and SPT. Thus, although there is clear cross-correlation between the signals, the cross-covariance between the CMB and cosmic shear auto-correlation function will still be relatively small. The reason is that the auto-correlation covariance contains reconstruction noise and shape noise, in addition to the auto-correlation signal. For the cross-covariance, only the signal component of the cross-correlation is relevant, and it is smaller than the auto-correlation.

2.4. Supernova data

For the evolving dark energy models, we furthermore use measurements of Type Ia supernovae from the Pantheon+ compilation (Scolnic et al. 2022), consisting of 1701 light curves of 1550 spectroscopically confirmed SNIa with redshifts z ∈ (0.001, 2.26). Inference of cosmological parameters from this dataset was presented in Brout et al. (2022).

2.5. Data combinations used

The data combinations used are summarised in Table 1. We focus on constraints from KiDS-Legacy alone, which we refer to as KiDS in the discussion and plots. The combination of SPT, Planck, and ACT is labelled CMB (this corresponds to the SPA likelihood in Camphuis et al. 2025). All probes that use the weak gravitational lensing effect, that is, CMB lensing, KiDS, and DES Y3, are collectively referred to as ‘lensing’. We label the low-redshift probes as low-z, that is, those that do not depend on the CMB. Lastly, we combine all probes but the supernovae into the low-z+CMB likelihood. All external likelihoods used in this work are publicly available and implemented in the COSMOSIS standard library6 (Zuntz et al. 2015).

Table 1.

Data combinations used in this analysis. The details are described in Sect. 2.

2.6. Inference

We use COSMOPIPE and sample the parameter space via the NAUTILUS7 sampler (Lange 2023) to generate samples from the posterior distribution. As a nested sampling method (Skilling 2006), NAUTILUS allows for direct computation of the evidence and hence easy estimation of the Bayes factor B for model comparison:

B = p ( D | M 1 ) p ( D | M 2 ) , Mathematical equation: $$ \begin{aligned} B = \frac{p(D|M_1)}{p(D|M_2)}\;, \end{aligned} $$(2)

where p(D|M1) is the evidence of the model M1. If B > 1, M1 is preferred over model M2 and vice versa for B < 1. The amount of preference can be quantified using the Jeffreys’ scale. It quantifies a model preference with |log10B|> [0.5, 1, 2] as ‘substantial’, ‘strong’, and ‘decisive’. The Bayes ratio suffers from several shortcomings: it is prior-dependent, not a probability, and, as a result, the Jeffreys scale is somewhat arbitrary. To avoid prior effects, we also report the tension between the two models based on suspiciousness (Handley & Lemos 2019), where the parameter posteriors indicate some level of tension. For more details, we refer to Section 3.2.1 in Stölzner et al. (2025). However, since most models will be very consistent with the fiducial ΛCDM analysis, we will use the Bayes factors. For NAUTILUS, we use the following settings: flive = 0.01, Nlive = 3000, and Neff = 10000.

We use the cosmological parameter inference code COSMOSIS (Zuntz et al. 2015) to facilitate all of our computations and use CAMB (Lewis et al. 2000) for the corresponding linear power spectrum to be fed into HMCODE2020 for the non-linear power spectra. The likelihood for each probe is assumed to be a multivariate Gaussian likelihood with the cosmic shear covariance matrix calculated via the ONECOVARIANCE8 (Reischke et al. 2025) at the ΛCDM best-fit cosmology.

A list of model parameters is provided in Table 2. For brevity, we show only nuisance parameters relevant to cosmic shear. The flat ΛCDM parameters are the reduced Hubble constant h = H0[km s−1Mpc−1]/100, the cold dark matter and baryon densities ωcdm = Ωcdmh2 and ωb = Ωbh2, the spectral index of the primordial power spectrum and its amplitude, which we parametrise by either the variance σ82 in spheres of 8 Mpc/h, or by the amplitude of the matter power spectrum As at a pivot scale k = 0.05 Mpc−1. Since cosmic shear is most sensitive to S 8 = σ 8 Ω m / 0.3 Mathematical equation: $ S_8 = \sigma_8\sqrt{\Omega_{\mathrm{m}}/0.3} $, we use S8 by default as the sampling parameter and derive the others. For the extended models, we use the sum of the neutrino masses ∑mν, the spatial curvature parameter defined in Eq. (5), and the w0, wa-parametrisation as in Eq. (3).

Table 2.

Model parameters and their priors grouped by the concordance model, extensions, and nuisance.

3. Constraints on extended models

In this section, we present the results for the extended cosmological models and for different combinations of datasets. For quick reference, the Bayes factors for model selection are shown in Table 3. The marginal mean and its corresponding 68% credible interval for all probe and model combinations for the main cosmological parameters are summarised in Table 4. We also show the p-values, calculated using the recipe discussed in Stölzner et al. (2025) (see Sect. 3.2.1), for the KiDS data alone for each case. This result differs slightly from Wright et al. (2025b), where the maximum a posteriori (MAP) probability was explicitly calculated: here, instead, we use the MAP from the posterior samples. As a result, the p-values presented here are an upper limit.

Table 3.

Bayes factors, B as defined in Eq. (2), for the extended models for the different probe combinations.

Here, we focus on the constraints on the parameters specific to the extended models studied and their interplay with S8. For more details, we show the full contours for all cosmological parameters and the feedback parameter in Appendix B. We find that the nuisance parameters, in particular the intrinsic alignment parameters, as well as the redshift calibration parameters, are prior-dominated and exhibit no cross-talk with extended models and probes. More precisely, the extended model parameters do not exhibit strong degeneracies with the nuisance parameters in the range allowed by the priors and the overall signal-to-noise of the cosmic shear measurement. Lastly, we note that constraints on ΩK and ∑mν from lensing alone are not competitive with those from the CMB or BAO, and in particular with their combination. However, the current comparison is not on equal footing, as cosmic shear is still in its golden years. Hence, this situation will improve as we move to Stage IV surveys (see Euclid Collaboration: Blanchard et al. 2020; Euclid Collaboration: Archidiacono et al. 2025), or forecasted constraints on curvature or neutrinos).

3.1. Neutrinos

The observation of neutrino oscillations necessitates that at least two of the three neutrino mass eigenstates have non-zero mass. Hence, their sum ∑mν > 0. In what follows, we express the neutrino mass in natural units, that is, c = 1. In the fiducial KiDS analysis, ∑mν = 0.06 eV was assumed, which is in line with the lowest allowed mass range in the normal hierarchy from oscillation measurements (de Salas et al. 2021; Esteban et al. 2020, 2024). Other laboratory experiments can instead give upper limits, ≤0.45 eV (at 90%) via double-β decay (this is, though, a weighted sum of the neutrino masses with the weights given by the PMNS mixing matrix; Katrin & Aker 2025). Due to their low mass and resulting high thermal velocities, neutrinos do not cluster on small scales, thus suppressing the growth of structures (see e.g. Lesgourgues & Pastor 2006; Tereno et al. 2009; Abazajian et al. 2015; Gerbino & Lattanzi 2017, for overviews as well as first constraints with cosmic shear). Constraints from cosmology are mainly dominated by the CMB (Planck Collaboration VI 2020; Camphuis et al. 2025), limiting ∑mν < 0.17 eV. In combination with BOSS (Ivanov et al. 2020) and DESI (Adame et al. 2025), the cosmological measurement of ∑mν approaches the lower bound set by oscillation experiments, assuming the normal hierarchy. Here, we investigate the influence of varying neutrino mass on the constraints on S8 by KiDS as well as the upper limits on ∑mν themselves. In the analysis presented here, we assume the normal hierarchy. For consistency with many other analyses, we run CAMB (Lewis et al. 2000) with fixed Neff and three neutrino species approximated as a single massive and two massless eigenstates (as in Planck Collaboration VI 2020) and pass their sum to HMCODE2020, a combination which was tested against HALOFIT in Camphuis et al. (2025) and found to give changes of 0.2σ. However, this implies the existence of two ultra-relativistic species. As a result, the inferred total neutrino mass tends to be systematically lower compared with the fully degenerate mass scenario (see e.g. Loureiro et al. 2019; Choudhury & Hannestad 2020).

Fig. 1 shows the marginalised constraints in the S8, ∑mν plane for the different probe combinations listed in Table 1. Since neutrinos affect clustering directly but also change the background by reducing Ωm when increasing ∑mν, the interpretation is less straightforward than, for example, for feedback. Interestingly, we find a small anti-correlation between S8 and ∑mν for KiDS alone. More probability mass at larger neutrino masses shifts the peak of the matter power spectrum towards lower wave numbers, k, thereby favouring slightly larger values of the spectral index, ns. This overall shift results in a slight amplitude gain for the cosmic shear signal, which, in turn, is accompanied by a slightly lower value of S8.

Thumbnail: Fig. 1. Refer to the following caption and surrounding text. Fig. 1.

Marginalised two-dimensional 68% and 95% credible regions in the S8, ∑mν plane. Different colours represent the probe combinations discussed in Table 1.

However, as this is not a significant shift in the parameters, the p-value at the MAP (S8 = 0.793 with neutrinos marginalised and S8 = 0.812 for the fiducial analysis) is equally good as the one for the ΛCDM case with a fixed neutrino mass. The upper limit on the neutrino mass from KiDS alone is

∑ m ν ≤ 1.97 eV ( KiDS at 95 % ) , Mathematical equation: $$ \begin{aligned} \sum m_\nu \le 1.97\,\mathrm{eV} \quad (\mathrm{KiDS} \;\mathrm{at} \;95\%)\;, \end{aligned} $$

that is, 95% of the marginal posterior probability in ∑mν lies below this value. Adding DES and, crucially, CMB lensing breaks the degeneracy with S8, as the CMB lensing signal is sensitive to larger scales. The low-redshift probes, in turn, mainly fix the matter density, thus not breaking the degeneracy. The strongest constraints come from the CMB, particularly when combined with DESI data, as increasing ∑mν increases the expansion rate at the time of CMB release. To keep the sound horizon consistent with the data, Ωm must increase. However, DESI’s lower Ωm thus gives very tight constraints on the neutrino masses for which we find ∑mν ≤ 0.048 eV (at 95%), consistent with Camphuis et al. (2025) and in tension with the neutrino oscillation measurements, ∑mν ≥ 0.058eV for the normal hierarchy. In particular, these values appear to reject the normal hierarchy at 2.1σ and over 3σ for the inverted hierarchy. It is noteworthy that this value is also lower than the fiducial value in the KiDS-Legacy analysis. However, since this is a small change relative to the overall sensitivity of the cosmic shear data, it will not affect the results of that analysis.

As expected, the neutrino sector is not very well constrained with KiDS data alone, but constraints from cosmic shear alone are of similar quality compared to the previous KiDS-1000’s 3 × 2pt analysis (Tröster et al. 2021).

3.2. Dark energy equation of state

We vary the dark energy equation of state parameter with the standard (CPL; Chevallier & Polarski 2001; Linder 2003) parametrisation:

w ( a ) = w 0 + w a ( 1 − a ) , Mathematical equation: $$ \begin{aligned} w(a) = w_0 +w_a(1-a), \end{aligned} $$(3)

with the uniform priors given in Table 2. We do not explicitly impose w < −1/3, which is required for accelerated expansion, in the prior. In this model, w0 = −1 and wa = 0 corresponds to the cosmological constant Λ. The cosmological background is modified so that:

E 2 ( a ) = Ω m a − 3 + ( 1 − Ω m ) exp ( − 3 ∫ 1 a d a ′ 1 + w ( a ′ ) a ′ ) , Mathematical equation: $$ \begin{aligned} E^2(a) = \Omega _{\rm m}a^{-3} + (1-\Omega _{\rm m})\,\mathrm{exp} \left({-3\int _1^a\mathrm{d} a^\prime \frac{1+w(a^\prime )}{a^\prime }}\right), \end{aligned} $$(4)

where the spatial curvature still vanishes. This parametrisation has attracted considerable attention recently, owing to the DESI results, which have measured w0 and wa to be discrepant with Λ. This, in turn, has sparked a debate. For example, that the BAO data measures w(z) = − 1 at the redshifts where the BAO has been measured (see e.g. Cortês & Liddle 2024; Efstathiou 2025), or whether prior volume effects play a role (e.g. Herold & Karwal 2025; Steinhardt et al. 2025; Bayat & Hertzberg 2025), if theoretical priors (e.g Lewis & Chamberlain 2025; Toomey et al. 2025; Caldwell & Linder 2025) can reconcile this tension, or if this hints towards non-minimally coupled gravity theories (Ye et al. 2025).

Here, we test the full w0, wa parametrisation and the subclass where wa = 0, these are called w0waCDM and wCDM respectively. For KiDS alone, we find the following constraints:

w 0 = − 1 . 1 − 0.5 + 0.5 ( K i D S f o r w 0 C D M a t 68 % ) , w 0 = − 1 . 0 − 0.7 + 0.7 ( K i D S f o r w 0 w a C D M a t 68 % ) , w a = − 1 . 3 − 2.0 + 1.9 ( K i D S f o r w 0 w a C D M a t 68 % ) , Mathematical equation: $$ \begin{aligned} \begin{split}&w_0 =-1.1^{+\,0.5}_{-\,0.5} \quad (\mathrm KiDS\; for\; w_0\mathrm CDM\;at \;68\%),\\&w_0 = -1.0^{+\,0.7}_{-\,0.7} \quad (\mathrm KiDS\; for\; w_0w_a\mathrm CDM\;at \;68\%),\\&w_a = -1.3^{+\,1.9}_{-\,2.0} \quad (\mathrm KiDS\; for\; w_0w_a\mathrm CDM\;at \;68\%), \end{split} \end{aligned} $$

consistent with ΛCDM and thus the fiducial analysis of KiDS. The dark energy equation of state alters the expansion function and, consequently, the growth of structure. The constraints obtained from weak lensing on the equation of state are determined by the effective equation of state, weff, at the mean redshift of the source distribution for each tomographic bin. Different models are then distinguished by the amount of structure growth from that point in time to today, and by what that point in time represents, that is, by the distance-redshift relationship. Increasing weff moves the source closer to the observer at fixed z, decreasing the lensing signal, requiring an increase in S8. At the same time, however, the amount of structure growth at a given z > 0 is increased by increasing weff if the amplitude of the matter power spectrum is kept fixed. The exact degeneracy between S8 and weff therefore depends on the redshift baseline of the survey in question and is difficult to disentangle since S8 depends on Ωm, which itself modifies the distance-redshift relation. Generally, however, one does find a strong positive correlation between S8 and weff. This is why weak lensing benefits significantly from the addition of external probes that fix either σ8 or Ωm.

In Fig. 2, we present the results for the w0, wa parametrisation with external probes, showing that the lensing and low-z probe combinations are consistent with ΛCDM. The addition of CMB data to the low-redshift probes (green contours) moves the best fit away from ΛCDM, since the CMB is able to break degeneracies in the DESI data with Ωm, making the low-z+CMB combination particularly strong. Crucially, as the green contour, although shifted away slightly from a cosmological constant, is consistent with ΛCDM, the addition of BAO data as measured by DESI is extremely important to move the contours away from ΛCDM. We remark that the prior range on h (see Table 2) is fairly tight. This choice is based on the ΛCDM analysis of KiDS-Legacy (Wright et al. 2025b). It does not restrict the analysis in ΛCDM when combining with the CMB (Stölzner et al. 2025). However, opening the parameter space for dynamical dark energy leads to a large posterior probability at low values of h and high values of Ωm, as can be seen in Fig. B.1d. This occurs since the CMB is mostly sensitive to the physical density ρm ∼ h2Ωm. Hence, when performing an analysis that includes SPA, we increase the prior range for h. In turn, this changes the Bayes factor as it depends on the prior volume, again motivating reporting the suspiciousness as well.

Thumbnail: Fig. 2. Refer to the following caption and surrounding text. Fig. 2.

Marginalised constraints on dynamical dark energy in the w0, wa parametrisation, Eq. (3). The black contour contains all cosmic shear probes considered in this work, and the red contour is independent of the CMB. The blue and green contours differ only in that the blue one excludes the DESI DR2 BAO. In purple, the Pantheon+ SN sample has been added. The dashed lines mark the ΛCDM values.

Since the lensing probes are consistent with ΛCDM and provide an excellent fit with a p-value of 0.3, the Bayes factor, B in Eq. (2), that is the evidence of the w0waCDM model over the evidence of ΛCDM given the same data, clearly prefers a cosmological constant over dynamical dark energy with B = 0.07. It is noteworthy that this is a stronger preference for ΛCDM than the Bayes factor suggests for the low-z+CMB combination, for which we find B = 2.73 in favour of w0waCDM. Especially adding SN from Pantheon+ removes the lower part of the posterior and moves it closer to ΛCDM; the preference for dynamical dark energy becomes smaller in this case. In terms of suspiciousness, we find tensions of 2.6σ and 0.4σ with respect to ΛCDM for the low-z+CMB and lensing, respectively. The 2.6σ suspiciousness tension is not the same as the 2.6σ difference of ΩK from zero. The former is related to the Bayes factor, while the latter is a significance in the posterior space. In fact, those significances for w0 and wa are 3.25σ and 3.4σ, respectively. Furthermore, we investigate the tension between our lensing dataset and the CMB+BAO. We use the Monte Carlo parameter-shift method introduced by Raveri et al. (2020) and find a tension of 0.5σ in the w0, wa plane, driven by the larger uncertainties for dynamical dark energy when using the lensing data. In general, we do not find any shifts larger than 2σ with this method.

3.3. Spatial curvature

In a universe with spatial curvature, the (comoving) angular diameter distance changes from being equal to the comoving distance, χ, to

f K ( χ ) = { K − 1 / 2 sin ( K 1 / 2 χ ) for K > 0 χ for K = 0 | K | − 1 / 2 sinh ( | K | 1 / 2 χ ) for K < 0 . Mathematical equation: $$ \begin{aligned} f_K(\chi ) =\left\{ \begin{array}{lc} K^{-1/2}\sin (K^{1/2}\chi )&\mathrm{for} \;K> 0 \\ \chi&\mathrm{for} \; K = 0\\ |K|^{-1/2}\sinh (|K|^{1/2}\chi )&\mathrm{for} \;K< 0 \end{array} \right.. \end{aligned} $$(5)

Due to the closure equation, the Hubble function is modified as

E ( a ) H ( a ) H 0 = Ω m a − 3 + ( 1 − Ω m − Ω K ) + Ω K a − 2 , Mathematical equation: $$ \begin{aligned} E(a) \frac{H(a)}{H_0} = \sqrt{\Omega _{\rm m}a^{-3} + (1-\Omega _{\rm m}-\Omega _K)+\Omega _Ka^{-2}}, \end{aligned} $$(6)

where ΩK = −(c/H0)2K is the spatial curvature parameter. Lastly, we assume that the modified structure formation in a spatially curved universe is entirely dictated by the linear power spectrum, while non-linear corrections are implicitly accounted for via the background. Terasawa et al. (2022) discuss the effect of ΩK ≠ 0 on the non-linear power spectrum using a separate universe approach and the halo model. The leading-order terms are captured via the linear growth in our approach, while higher-order terms are neglected. Hence, the results presented here can be seen as conservative.

The CMB (e.g. Planck Collaboration VI 2020; Camphuis et al. 2025) yields strong constraints on ΩK via the position of the first acoustic peak in the CMB angular power spectrum. Due to its degeneracy with the matter density, a combination with BAO data tightens constraints significantly, providing ΩK = (0.26 ± 0.11)×10−2 roughly 2σ away from spatial flatness.

Fig. 3 displays the constraints on ΩK together with S8 from KiDS, lensing, low-z, and all considered probes in black, red, blue, and green, respectively. The dashed line indicates a spatially flat universe, and ΩK is shown over the whole prior range. One can see that KiDS alone provides only mild constraints on ΩK, as it lies within the prior edge for ΩK > 0 and within the prior edge for ΩK < 0. This is expected, as measuring spatial curvature requires a meaningful observation of absolute physical scales, which lensing struggles to provide. Alternatively, adding CMB lensing already increases the constraining power substantially, as it provides measurements on larger scales and can therefore resolve the peak of the matter power spectrum, providing constraints on Ωm and hence ΩK, yielding

Ω K = 0.00 ± 0.04 ( lensing for ΩK Λ CDM at 68 % ) . Mathematical equation: $$ \begin{aligned} \Omega _K = 0.00\pm 0.04\quad (\mathrm {lensing\;for\;} \Omega _K\Lambda \mathrm {CDM\;at\;} 68\% ). \end{aligned} $$

Thumbnail: Fig. 3. Refer to the following caption and surrounding text. Fig. 3.

Constraints on ΩKCDM in the S8, ΩK plane. The colour scheme is the same as in Fig. 1. The low-z+CMB constraint (green) lies right next to the dashed line. The prior range is indicated by the grey bands.

Adding BAO to cosmic shear provides similar constraints, again via breaking the Ωm degeneracy. The influence of the constraints on S8 is negligible. Adding the CMB recovers the constraints provided in Camphuis et al. (2025) due to the high signal-to-noise ratio of the acoustic peak position.

3.4. Consistency of KiDS-Legacy

Stölzner et al. (2025) provided several internal consistency checks of the KiDS-Legacy dataset. Here, we also summarise the consistency when adding more datasets and expanding the parameter space.

First, Fig. 4 shows the constraints for Ωm, the spectral index ns and S8 for the three extended models studied here in comparison to the fiducial analysis from Wright et al. (2025b). It can be seen that the modes of the posterior marginals are very stable, with the largest shift occurring in S8 when varying the neutrino mass, as discussed already before. Furthermore, the constraints on S8 degrade when dynamical dark energy is included. This is because a more negative weff moves sources further away at a given redshift, requiring less matter clustering, leading to a positive correlation between weff and S8. We also measure the Kullback-Leibler divergence, DKL, between marginal posterior distributions p1 and p2, that is their relative entropy:

D KL ( p 1 , p 2 ) = ∫ d x p 1 ( x ) log p 1 ( x ) p 2 ( x ) . Mathematical equation: $$ \begin{aligned} D_{\rm KL}(p_1, p_2) = \int \mathrm{d} x\,p_1(x)\log \frac{p_1(x)}{p_2(x)}\;. \end{aligned} $$(7)

Thumbnail: Fig. 4. Refer to the following caption and surrounding text. Fig. 4.

Constraints on ΛCDM parameters from KiDS-Legacy alone when opening up the parameter space. The grey line in the marginalised Ωm indicates the prior range. All other parameters have hard priors as shown.

The largest Kullback-Leibler divergence in S8 is DKL = 0.45 in the case of the neutrinos, a shift of 1.2σ of the marginal mean relative to the fiducial analysis.

In Fig. 5, we show the consistency of different probe combinations in the Ωm, S8 plane and how dynamical dark energy moves these specific contours. One can see the shift to higher Ωm and hence S8 when adding the DESI results to the low-redshift and the CMB data (from the blue to the green contour). This shift, however, is reduced when adding in the SN from Pantheon+, bringing Ωm down again. This does not mean that the tension with Λ is reduced, as only the negative wa tail of the posterior is removed. While this moves the peak of the posterior closer to ΛCDM, the overall discrepancy with it stays the same (as was shown in Fig. 2). We observe the same shift for this data combination when moving from ΛCDM to w0waCDM (from the cyan to the purple contour). It is only in these scenarios where SN observations add constraining power. While we do not show it explicitly, low-z+SPA+Pantheon+ in ΛCDM agrees with low-z+SPA. Lastly, we show the effect on the lensing probes (from the black contour to the red contour) when using the w0, wa parametrisation, which is consistent with the fiducial analysis.

Thumbnail: Fig. 5. Refer to the following caption and surrounding text. Fig. 5.

Constraints on Ωm and S8 for different data combinations in the w0, wa parametrisation, Eq. (3). The red and the cyan contours assume a ΛCDM prior.

Fig. 6 displays a summary of the main probe and model combinations tested in this work. We show the marginal mean, together with the 68% confidence interval, of the main parameter constrained by cosmic shear, S8, for each combination, and highlight the fiducial ΛCDM analyses by KiDS-Legacy and Planck in blue and grey, respectively. Additionally, we present the case in which we keep wa fixed at zero and only vary w0, referred to as wCDM. We observe that the marginal constraints on S8 remain stable as the parameter space is expanded or additional probes are included. In ΛCDM, we find S8 to be unchanged and a per cent measurement from lensing alone, which turns into a sub-per cent measurement when combining all probes considered here. The shift in S8 when varying the neutrino mass reduces when including external data, as expected, as the combination of CMB and BAO pushes the neutrino mass very close to the value used in the fiducial analysis.

Thumbnail: Fig. 6. Refer to the following caption and surrounding text. Fig. 6.

Marginal mean of S8 and the 68% credible interval for all extended models and probe combinations tested in this work. The blue band shows the ΛCDM constraints from KiDS (Wright et al. 2025b), while the grey band shows the constraints from Planck (Planck Collaboration VI 2020).

For dynamical dark energy, wCDM does not affect the marginal mean of S8 strongly and simply increases the error bars. This is consistent with the fact that BAO and the CMB agree at w0 = −1 when wa is held fixed. However, when considering the full w0, wa parametrisation, the combination of all probes actually prefers high S8. The shift is mainly driven by an increasing Ωm, while σ8 does change less. This can, again, be quantified by the Kullback-Leibler divergence for which we find DKL = 3 (2.3σ) for S8 but DKL = 1.4 (1.1σ) for σ8. Interestingly, the value of σ8 is lowered in the dynamical dark energy scenario. However, as previously discussed, the preference for this extended model is relatively small.

As indicated in Table 4, some of the parameters are not constrained by the likelihood but rather via our prior choice; this is most pronounced for log10TAGN. Consequently, those posteriors have very non-Gaussian distributions, which can lead to projection effects. The main parameter of interest in the previous discussion, however, is S8 whose interplay with log10TAGN is studied in Appendix A. Since we do not find significant changes in S8 due to log10TAGN, we conclude that projection effects are not dominant.

Table 4.

Marginal constraints for all probe combinations (compare Table 1) and models tested in this work.

4. Summary

We present the extended cosmology analysis of the final cosmic shear catalogue of the Kilo-Degree Survey (KiDS-Legacy) in combination with external cosmic shear data from the Dark Energy Survey (DES), as well as external probes from the cosmic microwave background (CMB), baryon acoustic oscillations (BAOs), and redshift space distortion (RSD). Due to the consistency of KiDS-Legacy with other cosmological probes, especially in S 8 = σ 8 Ω m / 0.3 Mathematical equation: $ S_8 = \sigma_8\sqrt{\Omega_{\mathrm{m}}/0.3} $, we can, for the first time, reliably combine with all external datasets. We derive constraints on the sum of neutrino masses, the spatial curvature of the Universe, and the equation of state of dark energy. Furthermore, we investigate the robustness of the KiDS-Legacy data to different combinations of data and parameter space extensions.

We find that KiDS-Legacy prefers ΛCDM over extended models, with almost identical goodness of fit across all extended models considered, and that the Bayes factors are all below unity, disfavouring extended models. By combining KiDS-Legacy with DES Y3 and CMB lensing data from Planck, the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT), we define a lensing probe which enhances the consistency with ΛCDM, rejecting extended models according to the Jeffreys scale, especially time-evolving dark energy models, with a Bayes factor B < 1. The equation of state of these models is constrained to w 0 = − 0 . 92 − 0.42 + 0.45 Mathematical equation: $ w_0 = -0.92^{+0.45}_{-0.42} $ and w a = − 0 . 88 − 1.56 + 1.45 Mathematical equation: $ w_a = -0.88^{+1.45}_{-1.56} $ from weak gravitational lensing alone. Curvature and neutrinos are only poorly constrained by weak gravitational lensing with Ω K = 0 . 001 − 0.041 + 0.040 Mathematical equation: $ {\Omega_K = 0.001^{+0.040}_{-0.041}} $ and ∑mν ≤ 1.61 eV.

Adding clustering data from BAOs and RSD significantly improves constraints compared to those from weak gravitational lensing alone, as Ωm becomes tightly constrained, thereby breaking one of the main lensing degeneracies and allowing it to measure the amplitude of the linear matter power spectrum, σ8. This low-redshift, low-z probe collection is independent of the CMB and is, to a substantial degree, consistent with ΛCDM, with all constraints consistent with ΛCDM within 1σ, and hence all Bayes factors disfavour the extended models.

When combining low-redshift data with CMB primary anisotropies and CMB lensing, we recover constraints on the neutrino mass and curvature from the CMB alone, which are much more sensitive to these parameters. For dynamical dark energy, however, the different preferences for Ωm between the CMB and low-z probes shift w(a) away from ΛCDM by roughly 3σ if allowed to vary with redshift. This, however, is less strong in terms of model selection than the preference of lensing probes for ΛCDM. More quantitatively, the Bayes factor, B is 0.07 (0.02) using our lensing (low-z) probes in preference of w0waCDM over ΛCDM, thus corresponding to strong (very strong) evidence in favour of ΛCDM. In contrast, when combining all probes, we only find B = 2.73.

The cosmic shear analysis of KiDS-Legacy is robust to minimal extensions of ΛCDM, preferring it over all other models while maintaining statistically stable inferred values of S8 when the parameter space is expanded, or external probes are added. This extends the fiducial analysis of KiDS-Legacy naturally, while arriving at a similar conclusion: at the end of cosmic shear Stage-III surveys, cosmological probes, spanning 13 billion years, are strongly in favour of ΛCDM. With the tentative evidence for dynamical dark energy in DESI, cosmic shear is set to move to Stage-IV as well, providing much tighter constraints and decisively measuring w(z).

Acknowledgments

The authors would like to thank L. Balkenhol, E. Camphuis, and S. Galli for their help and for pointing us to the right places for constructing the CMB likelihood. RR, AD, HHi, and CM are supported by a European Research Council Consolidator Grant (No. 770935). AHW is supported by the Deutsches Zentrum für Luft- und Raumfahrt (DLR), under project 50QE2305, made possible by the Bundesministerium für Wirtschaft und Klimaschutz, and acknowledges funding from the German Science Foundation DFG, via the Collaborative Research Center SFB1491 “Cosmic Interacting Matters - From Source to Signal”. BS, and ZY acknowledge support from the Max Planck Society and the Alexander von Humboldt Foundation in the framework of the Max Planck-Humboldt Research Award endowed by the Federal Ministry of Education and Research. HHi is supported by a DFG Heisenberg grant (Hi 1495/5-1), the DFG Collaborative Research Centre SFB1491, and the DLR project 50QE2305. BJ acknowledges support from the ERC-selected UKRI Frontier Research Grant EP/Y03015X/1 and by STFC Consolidated Grant ST/V000780/1. MA is supported by the UK Science and Technology Facilities Council (STFC) under grant number ST/Y002652/1 and the Royal Society under grant numbers RGSR2222268 and ICAR1231094. LM acknowledges the financial contribution from the PRIN-MUR 2022 20227RNLY3 grant “The concordance cosmological model: stress-tests with galaxy clusters” supported by Next Generation EU and from the grant ASI n. 2024-10-HH.0 “Attività scientifiche per la missione Euclid – fase E”. AL acknowledges support from the Swedish National Space Agency (Rymdstyrelsen) under Career Grant Project Dnr 2024-00171. MvWK acknowledges the support by UK STFC (grant no. ST/X001075/1), the UK Space Agency (grant no. ST/X001997/1), and InnovateUK (grant no. TS/Y014693/1). MB is supported by the Polish National Science Center through grant no. 2020/38/E/ST9/00395. JHD acknowledges support from an STFC Ernest Rutherford Fellowship (project reference ST/S004858/1). BG acknowledges support from the UKRI Stephen Hawking Fellowship (grant reference EP/Y017137/1). CH acknowledges support from the Max Planck Society and the Alexander von Humboldt Foundation in the framework of the Max Planck-Humboldt Research Award endowed by the Federal Ministry of Education and Research, and the UK Science and Technology Facilities Council (STFC) under grant ST/V000594/1. CG is funded by the MICINN project PID2022-141079NB-C32. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya. SJ acknowledges the Dennis Sciama Fellowship at the University of Portsmouth and the Ramón y Cajal Fellowship from the Spanish Ministry of Science. KK acknowledges support from the Royal Society and Imperial College. LL is supported by the Austrian Science Fund (FWF) [ESP 357-N]. CM acknowledges support under the grant number PID2021-128338NB-I00 from the Spanish Ministry of Science. NRN acknowledges financial support from the National Science Foundation of China, Research Fund for Excellent International Scholars (grant n. 12150710511), and from the research grant from China Manned Space Project n. CMS-CSST-2021-A01. TT acknowledges funding from the Swiss National Science Foundation under the Ambizione project PZ00P2_193352 MY, SSL and HHo acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant agreement No. 101053992). Software: The figures in this work were created with MATPLOTLIB (Hunter 2007), making use of the NUMPY (Harris et al. 2020), SCIPY (Virtanen et al. 2020), COSMOPOWER (Spurio Mancini et al. 2022) and GETDIST (Lewis 2025) software packages. Kilo-Degree Survey: Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 179.A-2004, 177.A-3016, 177.A-3017, 177.A-3018, 298.A-5015. Dark Energy Spectroscopic Instrument: This research used data obtained with the Dark Energy Spectroscopic Instrument (DESI). DESI construction and operations are managed by the Lawrence Berkeley National Laboratory. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High-Energy Physics, under Contract No. DE–AC02–05CH11231, and by the National Energy Research Scientific Computing Centre, a DOE Office of Science User Facility under the same contract. Additional support for DESI was provided by the U.S. National Science Foundation (NSF), Division of Astronomical Sciences under Contract No. AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research Laboratory; the Science and Technology Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Science and Technology of Mexico (CONACYT); the Ministry of Science and Innovation of Spain (MICINN), and by the DESI Member Institutions: www.desi.lbl.gov/collaborating-institutions. The DESI collaboration is honoured to be permitted to conduct scientific research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the Tohono O’odham Nation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U.S. National Science Foundation, the U.S. Department of Energy, or any of the listed funding agencies. SDSS-IV: Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High Performance Computing at the University of Utah. The SDSS website is www.sdss4.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration, including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Centre for Astrophysics|Harvard & Smithsonian, the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional/MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. Dark Energy Survey: This project used public archival data from the Dark Energy Survey (DES). 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 FacilitiesCouncil of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center 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. Planck: Based on observations obtained with Planck (http://www.esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada. SPT: Supported by the National Science Foundation (NSF) through awards OPP-1852617 and OPP-2332483. Partial support is also provided by the Kavli Institute of Cosmological Physics at the University of Chicago. Argonne National Laboratory’s work was supported by the U.S. Department of Energy, Office of High Energy Physics, under contract DE-AC02-06CH11357. ACT: Support for ACT was through the U.S. National Science Foundation through awards AST-0408698, AST- 0965625, and AST-1440226 for the ACT project, as well as awards PHY-0355328, PHY-0855887 and PHY- 1214379. Funding was also provided by Princeton University, the University of Pennsylvania, and a Canada Foundation for Innovation (CFI) award to UBC. ACT operated in the Parque Astronómico Atacama in northern Chile under the auspices of the Agencia Nacional de Investigación y Desarrollo (ANID). Author contributions: All authors contributed to the development and writing of this paper. The authorship list is given in three groups: the lead authors (RR, BS, BJ) followed by two alphabetical groups. The first alphabetical group includes those who are key contributors to both the scientific analysis and the data products. The second group covers those who have either made a significant contribution to the data products or to the scientific analysis.

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5

To determine fractions, we use the template-fitting code BPZ (Benítez 2000), which provides TB as an output.

Appendix A: Feedback prior

The feedback parameter, log10TAGN, is barely constrained (compare Table 4). If at all, one can find an upper limit for log10TAGN for some probe combinations. Since large log10TAGN corresponds to stronger feedback, this is indicative of a ceiling to the amount of feedback. This is, of course, only true for the shape of the power spectrum suppression facilitated by log10TAGN and for the summary statistic used. Broxterman et al. (2025) for example, reconstruct the matter power spectrum using a different summary statistic, ξ±, and find a somewhat larger suppression than indicated by the posterior for log10TAGN.

Since the influence of S8 on the extensions studied here was already established in the main text, the main question we can still address is whether different prior choices on log10TAGN influence S8. To this end, we run four additional chains using only the KiDS-Legacy data, changing the prior range of log10TAGN to a wide prior: log10TAGN ∈ [6.8, 9.6], (compare to the fiducial prior adopted in Table 2: log10TAGN ∈ [7.3, 8.3]). This prior range allows the suppression of the matter power spectrum to exceed unity; that is, the power will be enhanced. This happens if log10TAGN < 7.3 and affects only scales with k > 1 h Mpc−1. However, the effect can reach 10% at k = 10 h Mpc−1. Furthermore, this broad prior is much wider than what HMCODE2020 has been calibrated on, so extrapolation might not give the most realistic results. However, we only want to show that we are not sensitive to that choice. Additionally, we use recent constraints on log10TAGN from Fast Radio Bursts (FRBs, Reischke & Hagstotz 2025) to provide a somewhat tighter prior, in particular favouring stronger feedback models over the KiDS data. These constraints are also consistent with previous analyses using KiDS-1000 and the thermal Sunyaev-Zel’dovich effect (Tröster et al. 2022).

The result of this exercise is shown in Fig. A.1, illustrating that S8 is very stable with respect to the prior choices. The tighter prior on log10TAGN increases S8 slightly and improves the constraints marginally. This is due to the modest preference for stronger feedback among the FRBs, which requires a larger S8. Lastly, we check the posterior of S8 for two other cases: firstly, for a fixed value of log10TAGN = 7.9, which corresponds to the value found in Reischke & Hagstotz (2025) and is consistent with the suppression measured in Bigwood et al. (2024), Kovač et al. (2025) and the fgas − 8σFLAMINGO simulation (Schaye et al. 2023; Schaller et al. 2025), that is fairly strong feedback. Secondly, we assume no feedback and infer S8 from a pure CDM power spectrum. The results are shown as a green and grey histogram, respectively. In Dark Energy Survey and Kilo-Degree Survey Collaboration (2023) it was already shown that COSEBIs within the angular scales used here are expected to be rather insensitive to feedback. As before, we observe that larger feedback shifts S8 slightly higher. The fact that we do not have to marginalise over the feedback results in marginally tighter constraints on S8.

Thumbnail: Fig. A.1. Refer to the following caption and surrounding text. Fig. A.1.

Marginal posterior distribution of S8 for different prior choices for the feedback parameter log10TAGN, only using KiDS-Legacy cosmic shear data.

In conclusion, this plot shows that the fiducial analysis of KiDS-Legacy and, in combination with the results shown here, also the extended cosmological analysis are not influenced by the prior on the feedback parameter when using COSEBIs as a summary statistic, in line with Wright et al. (2025b).

Appendix B: Full cosmological posteriors

In this Appendix, we show the posterior distributions of the cosmological parameters and the feedback parameter log10TAGN. Figs. B.1a to B.1d show the credible regions for the three extensions for KiDS-Legacy, lensing, low-z and low-z+CMB (compare Table 1), respectively.

Thumbnail: Fig. B.1. Refer to the following caption and surrounding text. Fig. B.1.

Marginal constraints for the full set of cosmological parameters and the feedback parameter for all probe combinations and model extensions discussed in this work.

Here, one can see a few trends discussed in the main text. Figs. B.1a to B.1b e.g. show that allowing the neutrino mass to vary leads to more probability mass at lower ns in Fig. B.1a, thus lowering S8 slightly. However, this degeneracy is broken by introducing any of the external probes studied here; large values of ns become excluded. In Fig. B.1b, one can see that in ΛCDM, gravitational lensing can constrain the Hubble constant slightly (black contour). This is possible because the range of scales tested by CMB lensing and cosmic shear is large enough to resolve the peak of the power spectrum and the transition from linear to non-linear scales, both of which depend on h.

Fig. B.1d shows the effect of opening up the parameter space to dynamical dark energy when using the CMB, as the strong degeneracy between Ωm is exposed in this case. In ΛCDM, this is not visible since the expansion history is fixed by the combination of the CMB and BAO, limiting the range of Ωm values. This allows the exploration of low values of h that are, in principle, rejected by our prior choices (see Table 2). For chains including the CMB and dynamical dark energy, however, we allow for a wider h prior to discover the full shape of the posterior as driven by the likelihood. Tröster et al. (2021) found a similar situation for ΩK; however, in this work, this is remedied by the inclusion of additional data, thereby removing the slight skewness towards negative ΩK in the CMB. In fact, we find consistent results with the analysis presented in Camphuis et al. (2025), with ΩK > 0 at 2.6σ. This, however, does not warrant a model preference as we show in Table 3. It is noteworthy that the addition of SNe from Pantheon+ blocks the high Ωm and low h part of the parameter space again.

All Tables

Table 1.

Data combinations used in this analysis. The details are described in Sect. 2.

In the text
Table 2.

Model parameters and their priors grouped by the concordance model, extensions, and nuisance.

In the text
Table 3.

Bayes factors, B as defined in Eq. (2), for the extended models for the different probe combinations.

In the text
Table 4.

Marginal constraints for all probe combinations (compare Table 1) and models tested in this work.

In the text

All Figures

Thumbnail: Fig. 1. Refer to the following caption and surrounding text. Fig. 1.

Marginalised two-dimensional 68% and 95% credible regions in the S8, ∑mν plane. Different colours represent the probe combinations discussed in Table 1.
In the text
Thumbnail: Fig. 2. Refer to the following caption and surrounding text. Fig. 2.

Marginalised constraints on dynamical dark energy in the w0, wa parametrisation, Eq. (3). The black contour contains all cosmic shear probes considered in this work, and the red contour is independent of the CMB. The blue and green contours differ only in that the blue one excludes the DESI DR2 BAO. In purple, the Pantheon+ SN sample has been added. The dashed lines mark the ΛCDM values.
In the text
Thumbnail: Fig. 3. Refer to the following caption and surrounding text. Fig. 3.

Constraints on ΩKCDM in the S8, ΩK plane. The colour scheme is the same as in Fig. 1. The low-z+CMB constraint (green) lies right next to the dashed line. The prior range is indicated by the grey bands.
In the text
Thumbnail: Fig. 4. Refer to the following caption and surrounding text. Fig. 4.

Constraints on ΛCDM parameters from KiDS-Legacy alone when opening up the parameter space. The grey line in the marginalised Ωm indicates the prior range. All other parameters have hard priors as shown.
In the text
Thumbnail: Fig. 5. Refer to the following caption and surrounding text. Fig. 5.

Constraints on Ωm and S8 for different data combinations in the w0, wa parametrisation, Eq. (3). The red and the cyan contours assume a ΛCDM prior.
In the text
Thumbnail: Fig. 6. Refer to the following caption and surrounding text. Fig. 6.

Marginal mean of S8 and the 68% credible interval for all extended models and probe combinations tested in this work. The blue band shows the ΛCDM constraints from KiDS (Wright et al. 2025b), while the grey band shows the constraints from Planck (Planck Collaboration VI 2020).
In the text
Thumbnail: Fig. A.1. Refer to the following caption and surrounding text. Fig. A.1.

Marginal posterior distribution of S8 for different prior choices for the feedback parameter log10TAGN, only using KiDS-Legacy cosmic shear data.
In the text
Thumbnail: Fig. B.1. Refer to the following caption and surrounding text. Fig. B.1.

Marginal constraints for the full set of cosmological parameters and the feedback parameter for all probe combinations and model extensions discussed in this work.
In the text

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