Robust extended-cosmology constraints from combined cosmic shear and CMB data: curvature-dependent non-linear modelling, neutrino-mass treatment, and joint covariances in KiDS-Legacy-era analyses

The KiDS-Legacy combined-probe analysis of Reischke et al. (A&A 709, A82, 2026) constrains spatial curvature ( $\Omega_K$ ), the total neutrino mass ( $\sum m_\nu$ ), and evolving dark energy ( $w_0w_a$ CDM) from cosmic shear combined with CMB, CMB lensing, BAO, RSD, and supernova data, but its inference rests on four modelling approximations: curvature-independent non-linear corrections to the matter power spectrum, a one-massive-plus-two-massless neutrino eigenstate treatment, a neglected shear–CMB cross-covariance, and prior-dependent Bayesian model selection. The problem is to replace each of these approximations with a consistent treatment — most importantly a curvature-dependent non-linear matter power spectrum — and to establish whether the extended-parameter constraints and the reported mild preference for evolving dark energy survive, thereby delivering corrected constraints with quantified modelling systematic uncertainty.

Background

Weak gravitational lensing (cosmic shear) measures the correlated distortions of distant galaxy shapes by intervening matter, constraining the amplitude of density fluctuations, conventionally summarized by $S_8=\sigma_8\sqrt{\Omega_\mathrm{m}/0.3}$ , where $\sigma_8$ is the rms of the linear density field in 8 Mpc/h spheres and $\Omega_\mathrm{m}$ the matter density fraction. The Kilo-Degree Survey (KiDS) completed its final data release (KiDS-Legacy); its two-point cosmic shear signal is compressed into COSEBIs (Complete Orthogonal Sets of E-/B-mode Integrals, a data-vector format that cleanly separates the lensing E-modes from B-mode systematics) in tomographic redshift bins. Predictions require the non-linear matter power spectrum $P_\mathrm{m,nl}(k,z)$ , for which the analysis uses HMCODE-2020 (Mead et al. 2021), a halo-model-based prescription calibrated on flat-geometry simulations, with a one-parameter ( $\log_{10}T_\mathrm{AGN}$ ) model of baryonic feedback from active galactic nuclei. KiDS-Legacy (Reischke et al. 2026) combines this shear signal, for the first time consistently with the CMB, with CMB primary anisotropies, CMB lensing (Planck, ACT, SPT), BAO (baryon acoustic oscillation) distances, RSD (redshift-space distortion) growth measurements, and the Pantheon+ compilation of type Ia supernovae, and constrains minimal extensions of the flat $\Lambda$ CDM model: spatial curvature $\Omega_K\neq0$ (where positive $\Omega_K$ denotes an open universe), a varying total neutrino mass $\sum m_\nu$ , and the CPL dark-energy parametrization $w(a)=w_0+w_a(1-a)$ with equation-of-state ratio $w=p/\rho$ . Neutrinos are treated in CAMB (a Boltzmann code computing cosmological power spectra) with fixed effective relativistic species number $N_\mathrm{eff}$ and three species approximated as one massive plus two massless eigenstates, as in Planck-era analyses. Model comparison uses Bayes factors $B$ (ratios of Bayesian evidences) graded on the Jeffreys scale, which are prior-volume dependent; the prior-insensitive suspiciousness statistic (Handley & Lemos 2019) is reported alongside. The final Stage-III (current-generation survey) result strongly favours $\Lambda$ CDM, with one exception: combining low-redshift probes with the CMB yields $B=2.73$ in favour of $w_0w_a$ CDM, a 2.6 $\sigma$ suspiciousness tension. A precursor analysis (Tröster et al. 2021, KiDS-1000 beyond-flat) shared the same modelling structure. Separate-universe simulations — which exploit the fact that a long-wavelength density or curvature perturbation acts on small-scale clustering like a change of background cosmology — allow the effect of $\Omega_K\neq0$ on the non-linear spectrum to be quantified without curved-box simulations; Terasawa et al. (2022, Phys. Rev. D 106, 083504) did this with the halo model, and Terasawa et al. (2023, arXiv:2310.13330) built an estimator that converts flat-geometry non-linear spectra into non-flat ones at percent-level accuracy, extendable to standard emulators. The KiDS-Legacy analysis instead propagates curvature only through the linear spectrum and the background, neglecting higher-order curvature terms; it also neglects the cross-covariance between the shear and CMB-lensing data vectors (argued to be small because the auto-covariances are noise-dominated).

Problem Statement

For the combined cosmic shear + CMB + low-redshift-probe inference of extended cosmologies as instantiated by KiDS-Legacy (Reischke et al. 2026), determine whether the constraints on spatial curvature $\Omega_K$ , total neutrino mass $\sum m_\nu$ , and evolving dark energy ( $w_0,w_a$ ), and the model-selection conclusions drawn from them, are robust to the analysis's own modelling approximations — that is, how the extended-parameter posteriors and the Bayes factors and suspiciousness respond to each of these approximations. Concretely: (1) treat the non-linear matter power spectrum in a curvature-consistent way (propagating the $\Omega_K\neq0$ dependence beyond the linear growth and background treatment) and re-derive the non-flat constraints; (2) quantify how the neutrino eigenstate approximation (one massive plus two massless species, fixed $N_\mathrm{eff}$ ) biases the inferred $\sum m_\nu$ relative to degenerate-mass and full-hierarchy treatments in the extended parameter spaces; (3) include the cross-covariance between the shear and CMB(-lensing) data vectors in the joint likelihood and assess its effect on combined constraints and evidence ratios; (4) establish whether the reported preference for $w_0w_a$ CDM over $\Lambda$ CDM in the low-redshift+CMB combination ( $B=2.73$ , 2.6 $\sigma$ suspiciousness) survives these modelling upgrades, or whether it can be attributed to the approximations; and assess the prior-robustness of these model-selection statements.

References

Reischke, R., Stölzner, B., Joachimi, B., et al., 'KiDS-Legacy: Constraining dark energy, neutrino mass, and curvature', A&A 709, A82 (2026). DOI: 10.1051/0004-6361/202558581, https://doi.org/10.1051/0004-6361/202558581 (arXiv:2512.11041, https://arxiv.org/abs/2512.11041)
Terasawa, R., Takahashi, R., Nishimichi, T., Takada, M., 'Separate universe approach to evaluate nonlinear matter power spectrum for nonflat $\Lambda$ CDM model', Phys. Rev. D 106, 083504 (2022). DOI: 10.1103/PhysRevD.106.083504, https://doi.org/10.1103/PhysRevD.106.083504 (arXiv:2205.10339, https://arxiv.org/abs/2205.10339)
Terasawa, R., Takahashi, R., Nishimichi, T., Takada, M., 'Flat to nonflat: Calculating nonlinear power spectra of biased tracers for nonflat $\Lambda$ CDM model' (2023). arXiv:2310.13330, https://arxiv.org/abs/2310.13330
Kou, R., Lewis, A., 'Improving CMB constraints on early Universe physics with LSS: A multi-probe forecast including cross-covariance', JCAP 08 (2025) 031. DOI: 10.1088/1475-7516/2025/08/031, https://doi.org/10.1088/1475-7516/2025/08/031 (arXiv:2504.13913, https://arxiv.org/abs/2504.13913)
Mead, A. J., Brieden, S., Tröster, T., Heymans, C., 'HMCode-2020: improved modelling of non-linear cosmological power spectra with baryonic feedback', MNRAS 502, 1401 (2021). DOI: 10.1093/mnras/stab082, https://doi.org/10.1093/mnras/stab082 (arXiv:2009.01858, https://arxiv.org/abs/2009.01858)
Loureiro, A., Cuceu, A., Abdalla, F. B., et al., 'Upper Bound of Neutrino Masses from Combined Cosmological Observations and Particle Physics Experiments', Phys. Rev. Lett. 123, 081301 (2019). DOI: 10.1103/PhysRevLett.123.081301, https://doi.org/10.1103/PhysRevLett.123.081301 (arXiv:1811.02578, https://arxiv.org/abs/1811.02578)
Choudhury, S. R., Hannestad, S., 'Updated results on neutrino mass and mass hierarchy from cosmology with Planck 2018 likelihoods', J. Cosmology Astropart. Phys. 2020 (7), 037 (2020). DOI: 10.1088/1475-7516/2020/07/037, https://doi.org/10.1088/1475-7516/2020/07/037 (arXiv:1907.12598, https://arxiv.org/abs/1907.12598)
Handley, W., Lemos, P., 'Quantifying dimensionality: Bayesian cosmological model complexities', Phys. Rev. D 100, 023512 (2019). DOI: 10.1103/PhysRevD.100.023512, https://doi.org/10.1103/PhysRevD.100.023512 (arXiv:1903.06682, https://arxiv.org/abs/1903.06682)
Du, G.-H., Li, T.-N., Liu, T., Zhang, J.-F., Zhang, X., 'Evidence for deviation in gravitational light deflection from general relativity at cosmological scales with KiDS-Legacy and CMB lensing', Sci. China-Phys. Mech. Astron. 69, 280411 (2026). DOI: 10.1007/s11433-026-2987-7, https://doi.org/10.1007/s11433-026-2987-7 (arXiv:2602.03110, https://arxiv.org/abs/2602.03110)
Lewis, A., 'CAMB v2: cosmological power spectra for high-precision surveys' (2026). arXiv:2607.14854, https://arxiv.org/abs/2607.14854
Tröster, T., Asgari, M., Blake, C., et al., 'KiDS-1000 Cosmology: constraints beyond flat $\Lambda$ CDM', A&A 649, A88 (2021). DOI: 10.1051/0004-6361/202039805, https://doi.org/10.1051/0004-6361/202039805 (arXiv:2010.16416, https://arxiv.org/abs/2010.16416)