The KV450 analysis (Hildebrandt et al. 2020), the first cosmic shear tomography survey combining optical and near-infrared imaging, found S8=0.737−0.036+0.040S_8 = 0.737^{+0.040}_{-0.036} from 450 deg2450\,\mathrm{deg}^2, in 2.3σ2.3\sigma tension with the Planck CMB analysis, and identified the accurate, unbiased calibration of tomographic source redshift distributions n(z)n(z) as its most important unresolved systematic: defensible alternative calibrations (spectroscopic or COSMOS-2015-based direct recalibration, clustering-zz, smoothed variants) move S8S_8 at the ∼0.6σ\sim 0.6\sigma level and are the only alternatives that alleviate the tension. Shear-calibration uncertainty (σm=0.02\sigma_m = 0.02), baryonic-feedback modelling, and the propagation of correlated, beyond-mean-shift n(z)n(z) errors add comparable weight to the systematic budget. The problem left open is whether tomographic n(z)n(z) can be calibrated and their full uncertainty propagated — jointly with the shear and baryonic systematics — so that cosmic shear constraints on S8S_8 are robust against the choice among defensible calibrations and the CMB-vs-shear discrepancy can be adjudicated as physical or systematic. Redshift calibration still moves S8S_8 at order 1σ1\sigma in the newest analyses and remains a leading candidate systematic for the survey-dependent S8S_8 pattern: the problem is open.

Background

Cosmic shear — the statistical distortion of distant galaxy shapes by gravitational lensing from the large-scale matter distribution — is one of the most powerful probes of the late-time universe. Because the lensing strength depends on distance, a shear measurement is interpreted through the redshift distribution n(z)n(z) of the source galaxies, and surveys split their sources into redshift bins ('tomography') to trace the growth of structure. Source redshifts in imaging surveys are photometric (photo-zz), estimated from multi-band fluxes rather than spectra; their per-galaxy errors scatter galaxies between bins, so the true ensemble n(z)n(z) of each tomographic bin must be calibrated externally. Within the standard flat Λ\LambdaCDM cosmology (cosmological constant plus cold dark matter), cosmic shear primarily constrains the amplitude parameter S8≡σ8Ωm/0.3S_8 \equiv \sigma_8\sqrt{\Omega_{\rm m}/0.3}, where σ8\sigma_8 is the rms matter-density fluctuation on 8 h−18\,h^{-1}Mpc scales and Ωm\Omega_{\rm m} the matter density fraction. The Kilo-Degree Survey combined with the VISTA Kilo-Degree Infrared Galaxy Survey (KiDS+VIKING-450, or KV450; Hildebrandt et al. 2020) was the first wide-field cosmic shear analysis with full nine-band optical+near-infrared photometry over 450 deg2450\,\mathrm{deg}^2, yielding S8=0.737−0.036+0.040S_8 = 0.737^{+0.040}_{-0.036} in a blind analysis, discrepant with the Planck-Legacy value for the cosmic microwave background (CMB) at 2.3σ2.3\sigma. KV450's fiducial redshift calibration was DIR — direct recalibration, in which a spectroscopic reference sample is re-weighted in magnitude space to match the survey sources. The paper compared this against a smoothed variant (sDIR), calibration against the deep 30-band COSMOS-2015 photo-zz catalogue, and clustering-zz estimators (CC and OQE, two published formulations of this cross-correlation approach) that infer n(z)n(z) from cross-correlation with spectroscopic galaxies. Each route carries distinct acknowledged failure modes: spectroscopic samples are incomplete and heterogeneous at the faint, high-redshift end where the shear sources live; COSMOS-2015 contains a non-negligible fraction of faint catastrophic outliers; clustering-zz requires galaxy-bias corrections and suffers noise and normalisation ambiguities. Switching among these defensible calibrations moved KV450's S8S_8 by ∼0.6σ\sim 0.6\sigma — and the redshift distribution was the only systematic whose alternatives alleviated the Planck tension. Two further systematics contribute comparable error-budget weight: the multiplicative shear-calibration bias mm of the shape-measurement pipeline, conservatively assigned σm=0.02\sigma_m = 0.02 on the basis of image simulations whose residual sensitivity traces uncertain assumptions about undetected faint galaxies and simulated photo-zz; and baryonic feedback from active galactic nuclei and star formation, which suppresses the non-linear matter power spectrum on small scales and was marginalised through the HMCode feedback amplitude BB (a halo-profile parameter in the HMCode non-linear power-spectrum model) constrained only by a prior informed by hydrodynamical simulations. Finally, KV450 propagated n(z)n(z) uncertainty only as per-bin mean-redshift shifts δzi\delta z_i, leaving bin-correlated errors and deformations of the n(z)n(z) shape largely unquantified, for which the paper called for deeper multi-line-of-sight spectroscopic samples and methodological development in estimating and propagating n(z)n(z) covariances. These issues are generic to all current and planned weak-lensing surveys (the Dark Energy Survey (DES), Hyper Suprime-Cam, Euclid, the Rubin Observatory Legacy Survey of Space and Time (LSST), and the Roman Space Telescope), which is why the paper's summary calls the robust determination of source redshift distributions 'one of the most challenging aspects for future cosmic shear surveys'.

Problem Statement

The problem left open by KV450 and inherited by every current cosmic shear survey is: can the tomographic source redshift distributions n(z)n(z) of a weak-lensing survey be calibrated, and their uncertainties propagated, well enough that the inferred S8S_8 is robust against the choice among defensible calibration methods — and, in particular, well enough to establish whether the ∼2\sim 2–3σ3\sigma discrepancy in S8S_8 between cosmic shear and CMB measurements is physical rather than a calibration artefact, or to establish that it cannot be so adjudicated? A complete answer must deliver: (i) calibrated n(z)n(z) for all tomographic bins of the source survey (KV450) — using the calibration families and published n(z)n(z) and covariance products of the cited analyses as inputs where available — whose residual bias and full uncertainty — including bin-correlated errors and deformations beyond simple mean-redshift shifts — are quantified and propagated into the cosmological posterior; and (ii) a demonstration on real survey data — assembling the calibration families and their S8S_8 constraints from the cited real-survey analyses — either that the scatter in S8S_8 among the calibration setups the community accepts as realistic — the families the source itself tests: direct spectroscopic recalibration, deep-photometric recalibration, clustering-zz, smoothed variants, and independent calibration samples — is driven below the statistical error, which it currently rivals in size, so that the CMB discrepancy is robust to the calibration choice; or that residual calibration freedom across defensible methods leaves the scatter at or above the statistical error, so that the tension cannot be adjudicated as physical. Shear-calibration and baryonic-feedback uncertainties enter the same likelihood as coupled context systematics — in KV450 the quantified systematic budget was dominated by the shear-bias uncertainty σm=0.02\sigma_m = 0.02, while baryon-feedback priors, intrinsic-alignment modelling and neutrino mass were found not to alleviate the Planck tension — but driving them below a dominance threshold is not a condition for settling the problem, which centres on n(z)n(z) calibration and the two-sided adjudication of the tension.

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