KiDS-1000: Detection of deviations from a purely cold dark matter power spectrum with tomographic weak gravitational lensing

Konrad Kuijken; Lucas Porth; Patrick Simon; Pierre Burger

REVIEW 4 major objections 5 minor 91 references

KiDS-1000: Detection of deviations from a purely cold dark matter power spectrum with tomographic weak gravitational lensing

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Tomographic weak lensing data reveal a redshift-dependent deviation from a purely cold dark matter power spectrum, with no detected structure growth between z≈0.7 and 0.4.

desk verdict A well-built z-resolved power-spectrum deprojection from KiDS-1000 with an intriguing but not-yet-robust no-growth signal; deserves peer review after the detection claim is tempered. read the letter →

arxiv 2502.04449 v2 pith:OZ44IFSS submitted 2025-02-06 astro-ph.CO

Patrick Simon , Lucas Porth , Pierre Burger , Konrad Kuijken This is my paper · ORCID

classification astro-ph.CO

keywords weakgravitationallensingcosmicshearmatterpowerspectrumstructuregrowthintrinsicalignmentsTikhonovregularizationS8tensionredshifttomography

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the three-dimensional matter power spectrum reconstructed from weak lensing data, with no analytic model for P(k,z), deviates from a purely cold dark matter prediction in a redshift-dependent way. Averaged over z≲1 the data are consistent with a low-amplitude CDM spectrum, but resolved into three redshift bins they show suppressed power at z≈0.3–0.6 and boosted power at z≈0.6–2, implying almost no measured structure growth between those epochs. If correct, this is a direct, model-free sign that non-linear structure growth is not purely CDM-like, unless hidden systematics in intrinsic alignment modeling or photometric redshifts are responsible. The paper develops a Tikhonov-regularized Bayesian deprojection to make such z-resolved reconstruction possible.

What carries the argument

The load-bearing object is the binned transfer function fδ(k,z), defined as the true matter power spectrum divided by a fiducial ΛCDM halofit reference; it is held constant inside cells of a 20×Nz grid in k and z. The projection from fδ cells to the tomographic shear correlations ξ±(ij)(θ) is a fixed linear matrix built from the lensing kernel, the calibrated source redshift distributions, and the NLA intrinsic-alignment model. A Tikhonov prior penalises squared differences between neighbouring k-bins in the same z-bin, suppressing the oscillatory noise that otherwise destabilises the deprojection; positivity priors and a Hamiltonian MCMC sample the 60-parameter posterior. The Tikhonov smoothing is what makes the three-bin redshift split statistically visible.

What would settle it

Fit the eNLA model with {A′IA,zpiv,η}={0.45,0.25,2.3} plus modest photo-z biases to the KiDS-1000 data vector and reconstruct fδ under the constant-AIA assumption; if the Z2–Z3 split disappears entirely under this systematic model, the power-spectrum anomaly is not needed to explain the data.

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Extended reading notes

Core claim

The central claim is that when the Kilo-Degree Survey lensing data are deprojected into binned values of fδ(k,z)=Pδ(k,z)/Pfid(k,z) with three redshift bins, the marginalised posterior gives a k-averaged f̄δ=1.15±0.28 in bin Z1=[0,0.3], f̄δ=0.57±0.27 in Z2=[0.3,0.6], and f̄δ=2.22±0.81 in Z3=[0.6,2]. Because the Z2 suppression and Z3 boost nearly cancel, a single-bin average looks close to ΛCDM; only the z-resolved view exposes the anomaly. In terms of the dimensionless power spectrum Δ²(k,z), the reconstructed spectra at z≈0.45 and z≈1.3 are statistically consistent with each other, so no growth is detected between z≈0.7 and 0.4, whereas growth is detected between z≈0.4 and 0.13. The authors list spurious systematics, intrinsic-alignment model inaccuracy, or delayed structure growth as possible causes; they demonstrate with fits that an evolving NLA amplitude plus moderate photo-z biases can reproduce the result.

Load-bearing premise

The reconstruction assumes the intrinsic-alignment model with a constant amplitude and the calibrated photometric-redshift distributions are accurate; if the real intrinsic-alignment amplitude varies with redshift or the photo-z distributions are biased, the apparent pause in structure growth could be an artifact.

Editorial extensions

If this is right

If the z-resolved anomaly is real, a purely CDM reference with Planck-level S8 requires a 20–30% suppression of power at k≈0.05–10 h/Mpc to match the lensing data, while a low-S8≈0.73 reference needs no suppression.
The reconstructed spectrum implies structure growth is concentrated at low redshift (z≈0.4 to 0.13), so probes weighting z≳0.7 would infer higher S8 than probes weighting z∼0.4, a redshift-dependent phrasing of the S8 tension.
N-body mock verification shows the pipeline recovers fδ=1 to about 10% accuracy under KiDS-1000-like noise, so the method transfers directly to larger surveys.
If the anomaly is instead an artifact, applying the same pipeline to future data with improved intrinsic-alignment and photo-z control should make the Z2–Z3 split shrink or vanish.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the split is a sharp test for baryonic feedback models, since matching it would require feedback to suppress power non-monotonically in redshift below z≈1, contrary to current simulation expectations.
Editorial inference: re-running this pipeline on the overlapping DES and HSC lensing surveys would settle whether the anomaly is survey-specific or a common signal; the authors note a related vanishing growth rate reported for KiDS-1000 and HSC Y3.
Editorial inference: a Stage IV application with more redshift bins should see the same deficit sharpen if it is cosmological, or disappear if it is caused by photo-z or intrinsic-alignment systematics that better calibration removes.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: this is the first model-free, z-resolved reconstruction of the matter power spectrum from KiDS-1000, and the method is genuinely well built. The Tikhonov-regularized deprojection with three redshift bins is new, and the verification on N-body mocks shows ~10% accuracy in the recovered fδ, which is solid. The null test for constant fδ rejects the baseline at p=0.012, so there is real evidence for some k- or z-dependence in the data.

The eye-catching result is the redshift split: fδ=0.57±0.27 in Z2 and 2.22±0.81 in Z3, meaning no growth between z≈0.7 and 0.4. That pattern is robust to the choice of reference, which is a point in its favor. But the paper itself, in Sect. 6, shows an eNLA intrinsic-alignment model plus moderate photo-z biases reproduces the KiDS-1000 split. The authors are honest about not being able to exclude either possibility. That is the load-bearing caveat. The claim is not yet established as cosmology.

There are two softer soft spots. First, the reference σ8 is tuned to the data so that the average fδ sits at one; that does not force the z-dependence, but it does mean the low-S8/consistent-with-CDM statement is partly built in. Second, the quantitative evidence against the null (Fig. 3) is computed with fixed projection parameters and does not marginalize over IA and photo-z uncertainty. The authors report those systematics add ~10% to errors, but the point estimate of the p-value would shift.

The title says 'Detection'; the body says 'could indicate... or potentially a non-standard model.' Those are in tension. A referee should ask for the claim to be tempered or for external data to break the degeneracy. That said, the methodology is careful, the mock tests are convincing, and the discussion of systematics is unusually complete. This deserves peer review and, after revision of the interpretation, publication.

Referee Report

4 major / 5 minor

Summary. The paper presents a Tikhonov-regularised Bayesian deprojection of KiDS-1000 tomographic shear correlations into a binned three-dimensional matter power spectrum, expressed as a transfer function fδ(k,z) relative to a ΛCDM reference. The authors verify the method on analytic mocks and on ray-traced N-body mocks with KiDS-1000-like noise, finding roughly 10% reconstruction accuracy. For KiDS-1000, averaging over z gives a spectrum consistent within 20% with a low-S8 reference, while a Planck-consistent reference requires 20–30% suppression on non-linear scales. Splitting into three redshift bins yields a suppressed middle bin (Z2: fδ=0.57±0.27) and a boosted high-z bin (Z3: fδ=2.22±0.81), which is interpreted as absence of detected structure growth between z≈0.7 and z≈0.4. The paper explicitly explores alternative explanations, including shear m-bias, photo-z bias, and a redshift-dependent intrinsic alignment amplitude, and concludes that an eNLA model plus moderate photo-z biases can reproduce the observed split.

Significance. If the z-resolved result is correct, it would be an important and surprising finding: a lensing-only reconstruction with minimal assumptions would indicate that non-linear structure growth between z≈0.7 and z≈0.4 differs from the pure-CDM expectation, possibly pointing to new physics or to unrecognised systematics. The methodological contribution is also valuable: the regularised deprojection with Hamiltonian Monte Carlo is carefully tested on both analytic and N-body mocks, and the paper is unusually transparent about the limitations and about systematic models that can mimic the signal. The stated goal of applying the technique to Stage IV surveys is well motivated. However, the central 'detection' of the z-split is not yet established, because the authors themselves provide a plausible systematic model (eNLA plus photo-z offsets) that reproduces the split and which they say they cannot exclude.

major comments (4)

[Sect. 5.1 and 5.2] The reference normalisation σ8=0.72 is fitted to KiDS-1000 to force ⟨fδ⟩≈1 (Table 1 note; Sect. 5.2: 'This way, the fδ,mn in our data are normalised to the average...'). Consequently, the statement that a low-S8 CDM model matches the average KiDS-1000 spectrum within 20% is partly by construction, and the Fig. 3 null test is not an independent test of that model. This does not affect the k- and z-dependent deviations, but the claim in the abstract that a low-S8 reference 'matches the KiDS-1000 spectrum within a 20% tolerance' should be phrased as a posterior fit, not as a finding from the reconstruction.
[Sect. 6 and Fig. 9] The authors demonstrate that an eNLA model with {A′IA,zpiv,η}={0.45,0.25,2.3} combined with photo-z offsets {δiz}={−0.017,−0.058,+0.027,+0.025,−0.043} reproduces the KiDS-1000 Z2/Z3 pattern, and they state they are 'unable to exclude either possibility.' Because the q-marginalisation in Sect. 3.5 varies only a constant AIA and photo-z within the Table 2 errors, this systematic class lies outside the error budget. The detection claim for no growth between z≈0.7 and z≈0.4 therefore requires excluding this class of model, which the paper does not do. At minimum, the null-test p-value in Fig. 3 should be recomputed while marginalising over such IA/photo-z systematics, or a quantitative prior should be placed on their allowed ranges.
[Sect. 5.4, 5.5, and Fig. A.5] The z-split is measured through medians of skewed, anti-correlated posterior distributions. The N-body verification itself shows a median bias in the direction of the effect: for fδ≡1 inputs at KiDS-1000 noise, the recovered medians are Z2≈0.81 and Z3≈1.29 (Sect. 5.5). The estimate that KiDS produces the observed split 'very roughly a 1/16 event, or less' (Sect. 6) is not a calibrated p-value and does not account for this known bias or for the anti-correlations shown in Fig. 5. A calibrated test statistic for the Z2−Z3 difference under the baseline, including the reconstruction bias, is needed to support the claim that the split is physical.
[Abstract and Sect. 6] The statement that the Z2/Z3 result is present 'regardless of the reference' is stronger than what is demonstrated. Section 6 shows that rescaling Pfidδ by a constant inside a z-bin leaves the inferred Δ2 invariant, but a reference with a different z-dependence of structure growth would redistribute fδ between the bins. The wording should be restricted to constant rescalings of the reference, or the invariance under more general reference changes should be established.

minor comments (5)

[Throughout] Powers of ten are often printed as inline numbers (e.g., '103', '104', '105') where the intended meaning is 10^3, 10^4, 10^5; this is confusing in equations and in the appendices.
[Fig. 3] The y-axis label 'C[-2log P|fδ] / per cent' is not self-explanatory; please define C and the null test statistic in the caption.
[Sect. 2.2] The notation J0,4(ℓθ) is used before J0 and J4 are defined in the text; define the Bessel functions at first use.
[Table 2] The table caption says 'in the Appendix' but Table 2 is in the main text; this should be corrected.
[Acknowledgements] The name 'Jeger Broxtermann' appears to be a typo for 'Jeger Broxterman'; please check the spelling.

Circularity Check

2 steps flagged · score 5.0 of 10

Low-S8 reference match is built into the sigma8=0.72 calibration; the z-resolved no-growth split retains independent content, so circularity is only partial.

fitted input called prediction [Table 1 note; Sect. 5.1; Abstract]
"† this value has been lowered from the 0.76 in Heymans et al. (2021), Table C.1, to obtain an average of ¯fδ≈ 1 over all k and z (Sect. 5.1) ... Conversely, a reference with a lower S8≈0.73 avoids suppression and matches the KiDS-1000 spectrum within a 20% tolerance."

The parameter σ8=0.72 is not inferred from an independent probe; it is lowered until the KiDS-1000 data give ⟨fδ⟩≈1. The abstract's statement that a low-S8 CDM reference 'avoids suppression and matches KiDS-1000 within a 20% tolerance' is therefore the defining calibration of the reference, not an empirical prediction. The average match is tautological; only the k- and z-dependence of fδ carries independent information.
self definitional [Sect. 5.2 and Sect. 5.4]
"In fact, the peak at ¯f0≈ 1 was our deliberate choice for KiDS-1000, achieved by lowering σ8 for Pfidδ(k,z) to 0.72 compared to the best-fitting value of 0.78 in Heymans et al. (2021) in order to move the peak from ¯f0≈ 0.9 to its final location ¯f0≈ 1. This way, the fδ,mn in our data are normalised to the average ⟨ fδ,mn⟩mn ≈ 1. The signal suppression in the middle redshift bin and the boost in the highest bin cancel each other ... resulting in the fδ = 0.99± 0.20 in the left panel."

The Nz=1 average fδ=0.99±0.20 is the identity of the chosen normalization: σ8 was adjusted so that the constant-fδ average equals 1, so reporting '0.99±0.20' restates the fit rather than measuring an amplitude. The z-split values (0.57±0.27 in Z2, 2.22±0.81 in Z3) are not fixed by this single amplitude and retain independent content, so the circularity is only partial.

full rationale

The deprojection pipeline itself is largely self-contained: Eq. (14) is a linear projection, the Tikhonov prior is explicitly stated, and the code is tested against analytic mock data generated with an independent code and against ray-traced N-body KiDS-1000-like mocks (Sect. 5.5). Those verification tests are not circular. The main circular component is the reference normalization: σ8=0.72 is deliberately lowered from 0.76 so that ⟨fδ⟩≈1, making the abstract's 'lower S8≈0.73 avoids suppression' and the Nz=1 fδ=0.99±0.20 calibrated inputs rather than outputs. The genuinely new claim, the Z2/Z3 split (fδ=0.57±0.27 and 2.22±0.81) implying little growth between z≈0.7 and 0.4, is not forced by that one-amplitude fit and is supported by mock-based accuracy tests (the 1/16 event statement in Sect. 6). The NLA constant-AIA and photo-z inputs are load-bearing assumptions, but the authors stress-test them: Sect. 6 shows an eNLA plus moderate photo-z bias model reproduces the KiDS-1000 split and states 'we are unable to exclude either possibility.' That is an acknowledged degeneracy/robustness caveat, not a circular reduction. Self-citations to Simon (2012) and Asgari et al. (2021) are not decisive in a circular way: the S12 method is re-derived and verified, while the Asgari/Heymans constraints are marginalised over and partly anchored to external spectroscopic data. Overall, circularity is partial: one fitted reference parameter is presented as a matching result, while the central z-dependent deviation has independent content.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new entities. Its free parameters are mostly analysis choices and the fitted reference normalization. The key assumptions are the standard lensing kernel, NLA intrinsic alignment model, halofit reference, Gaussian covariance, and photo-z calibration; the NLA and photo-z assumptions are load-bearing because the main z-split can be mimicked by their failure.

free parameters (5)

σ8 (reference normalization) = 0.72
Lowered from 0.76 of Heymans et al. (2021) to make the average fδ≈1 over all k and z (Sect 5.1). This normalizes the reference to the data.
Tikhonov regularization strength τ = 5
Chosen by hand after tests as a compromise between statistical precision and smoothing out real oscillations (Sect 3.2, Fig 2).
fδ,max (positivity prior bound) = 100
Wide upper bound for the top-hat prior on fδ, chosen to be uninformative (Sect 3.3).
σf (soft edge width) = 0.01
Softening of the positivity prior edges for numerical convenience of the HMC gradients (Sect 3.3).
Mesh choice (z-bin edges and k-range)
z-bins [0,0.3,0.6,2] and k range [0.01,20] h/Mpc with Nk=20 are analysis choices that affect the z-weighting and resolution (Sect 2.4, 5.4).

assumptions (7)

domain assumption Hybrid extended Limber approximation for the shear projection (Eq. 6)
Assumed accurate for the scales and redshifts used; standard in cosmic shear analyses.
domain assumption NLA model for intrinsic alignments with constant amplitude AIA (Eqs. 9-10)
The paper assumes the NLA model is accurate; Sect 6 shows a z-dependent eNLA model plus photo-z bias can reproduce the main result, so this assumption is load-bearing.
domain assumption Halofit (Takahashi et al. 2012) as the reference non-linear power spectrum
The reference Pfid is computed with halofit; deviations are measured relative to this model.
domain assumption Gaussian likelihood with covariance from Joachimi et al. (2021) Appendix E
The covariance accounts for shape noise, cosmic variance, super-sample covariance, and shear calibration uncertainty; assumes this model is correct.
domain assumption Source redshift distributions p_z(z) and their uncertainties from Hildebrandt et al. (2021)
The lensing kernel depends on the p_z; biases in these distributions are a potential systematic, as tested in Sect 6.
domain assumption Flat ΛCDM background cosmology with parameters from Heymans et al. (2021)
Used for distances and growth in the projection kernel; the paper marginalizes over uncertainties in Ωm and AIA.
ad hoc to paper Tikhonov smoothness prior on fδ along k
Introduced to stabilize the deprojection (Sect 3.2); it biases the reconstruction towards smooth k-dependence and must be accounted for when interpreting the results.

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Cite this review

Pith. "Pith review of KiDS-1000: Detection of deviations from a purely cold dark matter power spectrum with tomographic weak gravitational lensing." pith.science (2026). https://pith.science/paper/OZ44IFSS

@misc{pith2026250204449,
  author       = {Pith},
  title        = {Pith review of: KiDS-1000: Detection of deviations from a purely cold dark matter power spectrum with tomographic weak gravitational lensing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OZ44IFSS}},
  note         = {Machine review of arXiv:2502.04449}
}

abstract

Model uncertainties in the non-linear structure growth limit current probes of cosmological parameters. To shed more light on the physics of non-linear scales, we reconstructed the finely binned three-dimensional power-spectrum from lensing data of the Kilo-Degree Survey (KiDS), relying solely on the background cosmology, the source redshift distributions, and the intrinsic alignment (IA) amplitude of sources (and their uncertainties). The adopted Tikhonov regularisation stabilises the deprojection, enabling a Bayesian reconstruction in separate $z$-bins. Following a detailed description of the algorithm and performance tests with mock data, we present our results for the power spectrum as relative deviations from a $\Lambda\rm CDM$ reference spectrum that includes only structure growth by cold dark matter. Averaged over the full range $z\lesssim1$, a \emph{Planck}-consistent reference then requires a significant suppression on non-linear scales, $k=0.05$--$10\,h\,\rm Mpc^{-1}$, of up to $20\%$--$30\%$ to match KiDS-1000 ($68\%$ credible interval, CI). Conversely, a reference with a lower $S_8\approx0.73$ avoids suppression and matches the KiDS-1000 spectrum within a $20\%$ tolerance. When resolved into three $z$-bins, however, and regardless of the reference, we detect structure growth only in the range $z\approx0.4$--$0.13$, but not in the range $z\approx0.7$--$0.4$. This could indicate spurious systematic errors in KiDS-1000, inaccuracies in the intrinsic alignment (IA) model, or potentially a non-standard cosmological model with delayed structure growth. In the near future, analysing data from Stage IV surveys with our algorithm promises a substantially more precise reconstruction of the power spectrum.

Figures

Figures reproduced from arXiv: 2502.04449 by the authors.

**Figure 1.** Probability density distribution functions, p (i) z (z), of KiDS-1000 source galaxies within the five tomographic redshift bins (from i = 1 to i = 5): (0.1, 0.3], (0.3, 0.5], (0.5, 0.7], (0.7, 0.9], and (0.9, 1.2]. These estimates are from Hildebrandt et al. (2021). for a range of wave numbers and redshifts. The δD(x) denotes the Dirac delta function, and δ˜m(k,z) is the Fourier coefficient of a fluctuation mode at … view at source ↗

**Figure 2.** Impact of the Tikhonov regularisation used to suppress oscillating solutions fδ(k,z). Shown are, for a noise-free mock data vector and the KiDS-1000 error covariance, the posterior constraints (68% credible regions) on fδ averaged over the redshift bin Z1 = [0, 0.3] (left), Z2 = [0.3, 0.6] (middle), and Z3 = [0.6, 2] (right) with and without regularisation (dark orange τ = 5.0 with median as dashed line or light ora… view at source ↗

**Figure 3.** Test of data against a null model with identical fδ,mn = ¯f0 for all m and n (Nz = 3). Shown on the y-axis is the probability of −2 ln P(fδ,mn = ¯f0|d) being greater than in the null model. The dotted line is the result for one noisy verification data vector that has ¯f0 = 1; the solid line is for the KiDS-1000 data. The set-up in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Reconstructed matter power spectrum in KiDS-1000 for τ = 5.0 and Nk = 20 in three variants. The errors marginalise over uncertainties in the lensing kernel and IA. Shown are the posterior 68% CI about the median (lines). Left panel: Transfer function fδ(k,z) averaged o…

**Figure 5.** Figure 5: Correlation matrix of fδ,mn errors for the KiDS-1000 analysis with three redshift bins, corresponding to the middle and right panels in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Verification test of the analysis code for fδ(k,z) using: an Nbody simulated data vector, averaged over n = 1944 realisations of the KiDS-1000 data without IA; the code and data set-up for the KiDS1000 analysis has Nk = 20 k-bins and three redshift bins Z1 = [0, 0.3]…

**Figure 7.** Figure 7: Percentage fraction in total marginal posterior error of fδ,mn due to uncertainties in the lensing kernel and IA parameter. Shown is the statistical error relative to the full marginal error (RMS variance of posterior) as a function of k in three redshift bins without …

**Figure 8.** Figure 8: Power spectrum ratio ∆ 2 KiDS(k,z)/∆ 2 Planck(k,z) at z = 0.5. Shown in grey are the 68% and 95% CI from our analysis. The KiDS-1000 spectrum ∆ 2 KiDS(k,z) uses the fδ,mn in [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: KiDS-1000 results for fδ,mn, as in the middle panel of [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

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