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REVIEW 4 major objections 5 minor 70 references

Late-time suppression of structure growth as a solution for the $S_8$ tension

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that rapid late-time suppression of structure growth fits combined cosmic shear and CMB data better than LambdaCDM, hinting at a possible ~2 sigma deviation from General Relativity.

desk verdict A careful but over-optimistic model comparison: the DETG preference for late-time growth suppression is real at a modest level but likely inflated by fixing shear nuisance parameters to the LambdaCDM MAP. read the letter →

arxiv 2505.09176 v1 pith:ZTPFPT6C submitted 2025-05-14 astro-ph.CO

Ryo Terasawa , Masahiro Takada , Toshiki Kurita , Sunao Sugiyama This is my paper · ORCID
classification astro-ph.CO
keywords S8tensioncosmicshearmodifiedgravitystructuregrowthsuppressionDETGmodelindexCMBlensingbaryonicfeedback
topics Dark Energy
open problems Dark Energy
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

The paper asks whether the long-standing mismatch between the low clustering amplitude $S_8$ seen in cosmic shear and the higher value inferred from the early universe can be explained by suppression of structure growth at late times. It compares four ways to produce that suppression: baryonic feedback, massive neutrinos, a modified growth index $\gamma$, and a two-parameter 'dark energy tracking growth' (DETG) model in which growth slows as dark energy comes to dominate. Fitting the combined HSC Year 3 cosmic shear, Planck CMB, ACT DR6 CMB lensing, and DESI BAO data, the DETG models with the most rapid late-time suppression ($p=3,4,5,6$) improve the fit relative to $\Lambda$CDM at about the $2\sigma$ level with $\beta>0$, while baryonic and neutrino models do not. A sympathetic reader would care because this is a concrete, parameterized way to reconcile the $S_8$ tension with a genuine modification of gravity rather than a nuisance effect.

What carries the argument

The load-bearing object is the DETG rescaling of the linear matter power spectrum, which enters every cosmological probe through the growth factor. The model rescales the linear matter power spectrum at each redshift by $\alpha(z)=1-\beta[\Omega_{\rm DE}(z)/\Omega_{\rm DE}(0)]^p$, where $\beta$ sets the present-day suppression amplitude and $p$ sets how rapidly the suppression switches on as dark energy becomes dynamically important. This scale-independent rescaling is fed through the HMCODE16 nonlinear mapping so that cosmic shear and CMB lensing predictions inherit a distinct redshift dependence that the data can separate from baryonic feedback and neutrino free-streaming.

What would settle it

Refit the DETG models with the HSC nuisance parameters left free instead of fixed to the LambdaCDM maximum a posteriori values; if the posterior on $\beta$ returns to $\beta=0$ or the improvement over LambdaCDM disappears, the claimed ~2 $\sigma$ deviation from General Relativity is not robust.

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Extended reading notes

Core claim

The central claim is that the data prefer a scale-independent but redshift-dependent suppression of the linear growth factor that turns on only as dark energy begins to dominate, and that the suppression must evolve quickly toward the present day. The DETG model encodes this as $\alpha(z)=[D_{\rm DETG}(z)/D_{\Lambda\rm CDM}(z)]^2 = 1-\beta[\Omega_{\rm DE}(z)/\Omega_{\rm DE}(0)]^p$, with $\beta=0$ recovering $\Lambda$CDM. For $p=3,4,5,6$ the posterior favors $\beta>0$ with an improvement of $\Delta\chi^2/\Delta N_p\gtrsim 4$, which the authors read as a potential deviation from General Relativity at roughly $2\sigma$; the $\gamma$ growth-index model and the DETG model with $p=1,2$ are less preferred, and neither baryonic feedback nor varying neutrino mass resolves the tension.

Load-bearing premise

The analysis fixes the HSC cosmic shear nuisance parameters, including intrinsic alignment, photometric redshift errors, and shear calibration, to the values fitted for LambdaCDM in Li et al., so if the true cosmology has strong late-time growth suppression, those nuisance parameters would shift and the reported preference for $\beta > 0$ could be an artifact.

Editorial extensions

If this is right

  • If the DETG interpretation is right, the $S_8$ tension is not a systematic error in cosmic shear but a real signal of modified gravity active only once dark energy dominates.
  • The preferred parameter region ($p=3\text{--}6$, $\beta>0$) predicts growth suppression that strengthens toward $z=0$, which can be checked with redshift-resolved growth measurements from galaxy clustering and CMB lensing cross-correlations.
  • Models with slower suppression, such as the $\gamma$ model and DETG with $p=1,2$, fit markedly worse, so future data should discriminate the shape of the suppression rather than just its overall amplitude.
  • Because baryonic feedback and massive neutrinos do not produce the right redshift dependence, the analysis points away from within-$\Lambda$CDM fixes and toward genuine late-time physics.

Reading between the lines

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

  • The reported ~2 sigma preference depends on fixing the HSC nuisance parameters to the LambdaCDM maximum a posteriori values; if those parameters were allowed to float under modified-growth cosmologies, the central value of $\beta$ could shift and the significance could drop.
  • The analysis assumes the datasets are independent and ignores cross-covariance between HSC shear, Planck, and ACT; including cross-terms could either weaken or strengthen the preference.
  • A clean discriminator would be a growth-rate measurement at $z\sim 1$ to $2$: DETG with $p=5$ predicts nearly LambdaCDM growth there, while the $\gamma$ model with $\gamma=0.65$ predicts suppression already underway, so a high-redshift null result would separate the models.
  • If the preference survives in next-generation lensing surveys, it would motivate physical modified-gravity theories whose growth suppression tracks the dark-energy density rather than a purely empirical fit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper investigates whether late-time suppression of structure growth can resolve the S8 tension. It jointly analyzes HSC-Y3 cosmic shear two-point correlation functions, Planck-2018 primary CMB, ACT DR6 CMB lensing, and DESI Y1 BAO, comparing flat LambdaCDM with several extensions: baryonic feedback (HMCODE16 and HMCODE20), massive neutrinos, the gamma growth index model, and the dark-energy-tracking-growth (DETG) model of Lin et al. The central result is that the DETG model with p = 3, 4, 5, or 6 improves the chi-squared per added parameter by roughly 4-6 relative to LambdaCDM, corresponding to a claimed ~2 sigma preference for beta > 0, while baryonic and neutrino extensions are not favored. The paper concludes that models with more rapid late-time growth suppression offer a better combined fit, but it tempers this with the modest Bayes factors and the phenomenological nature of the ansatz.

Significance. If the result holds, it would provide evidence that a redshift-dependent suppression of structure growth, triggered by dark-energy domination, can simultaneously satisfy early- and late-universe data better than LambdaCDM or baryonic/neutrino alternatives. The analysis is technically careful: it uses public likelihoods, a consistent set of scale cuts, and a uniform treatment of the datasets across models. It also honestly reports the Bayes factors, which are considerably less favorable than the chi-squared differences. However, because the central claim rests on a fixed-nuisance comparison and a selection over the discrete parameter p, the significance is conditional on these choices. The paper is a useful exploratory step but does not yet establish a robust detection.

major comments (4)
  1. [III.B, Table II] The cosmic shear nuisance parameters (intrinsic alignment amplitude, photometric redshift shifts, shear calibration) are fixed to the MAP values from the LambdaCDM analysis of Li et al. This is a load-bearing assumption for the central claim. The DETG model modifies the growth factor and predominantly reduces the predicted cosmic shear 2PCFs at the scales used; a similar reduction can be produced by shifting these nuisance parameters. By fixing them to LambdaCDM MAP values, the reported Delta chi^2 values for DETG (p = 3-6) and the resulting '~2 sigma' statement in Section V are not a robust model comparison. The authors should marginalize over at least the intrinsic alignment and photo-z shift parameters for each extended model, or demonstrate that the preference for beta > 0 is insensitive to reasonable variations of these nuisances.
  2. [IV.D, Table II] The DETG model is tested for six discrete values of p (1 through 6), and the conclusion of a '~2 sigma deviation' is based on the subset p = 3, 4, 5, 6 that shows the largest chi-squared improvements. This multiple-testing selection is not accounted for; the probability of finding at least one p with such an improvement by chance is not evaluated. The Bayes factors in the same table (ln R between 0.3 and 1.1, categorized as 'barely worth mentioning' by the authors themselves) do not support a strong preference. The paper should either treat p as a free parameter with a prior, apply a trials factor, or de-emphasize the chi-squared-based significance in favor of the Bayes-factor evidence.
  3. [III.B] The analysis assumes the datasets are independent and ignores all cross-covariances (Section III.B: 'We assume the different datasets are independent and ignore the cross-covariance between them'). The ACT DR6 CMB lensing map overlaps substantially with the HSC-Y3 sky area, and the Planck CMB lensing signal is physically correlated with the HSC shear field. This approximation can bias the chi-squared differences used to rank models, especially at the 2-sigma level. The authors should at least assess the sensitivity by removing one dataset at a time or by adding a crude cross-correlation term, and discuss the expected impact on the Delta chi^2 values.
  4. [II.B.2, Eq. (6)] The DETG ansatz forces alpha(z) = 1 in the matter-dominated regime by construction, so any growth modification before dark-energy domination is excluded a priori. This is a strong model assumption that shapes the interpretation of the result: the data preference for beta > 0 is only a preference for a specific 'dark-energy-triggered' redshift dependence, not for growth suppression in general. The paper should explicitly acknowledge that the comparison with the gamma-index model does not cover models with earlier-onset suppression, and that the conclusion is limited to the chosen ansatz.
minor comments (5)
  1. [Abstract, Section I, Section II.B.1] There are typographical issues: 'within the the LambdaCDM framework' appears in the abstract and the introduction, and 'In this model, The linear growth rate' in Section II.B.1 should have a lowercase 'the'.
  2. [Figure 3] The caption describes the diagonal panels as 'lower-left' for xi+ and 'upper-right' for xi-, which is confusing given the panel layout. Please clarify which panels correspond to which combination, or reorder the panels with clear subfigure labels.
  3. [III.B and Table I] The nuisance parameters fixed to the Li et al. MAP values (intrinsic alignment amplitude, photometric redshift parameters, shear calibration) are not listed in Table I or defined in the text. For reproducibility, provide the full set of fixed values or a precise reference to the table in Li et al. where they are given.
  4. [IV.D, Section V] The statement that Delta chi^2 / Delta N_p of about 4-6 corresponds to 'more than a 2 sigma level under a naive Gaussian-like consideration' is not a standard significance statistic; the chi-squared difference should be interpreted with the effective number of degrees of freedom, and the Wilks or AIC/BIC criteria would be more appropriate. Please either justify the significance using a proper distribution or present it as a heuristic only.
  5. [II.C, II.B.2] HMCODE16 is used to compute the nonlinear matter power spectrum for the DETG model, although the model modifies the linear growth factor. This is an approximation; the paper should mention that the halo-model calibration has not been validated for DETG perturbations and discuss the possible systematic uncertainty on the chi-squared results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper performs parameter inference with external phenomenological models, and the improved fit for beta > 0 is a fitted result, not a prediction derived from its own inputs.

full rationale

The paper is a likelihood-based model comparison, not a derivation from first principles. The DETG and gamma models are external phenomenological ansatze taken from Lin et al. (2024) and Linder (2005), with amplitude parameters beta and gamma treated as free and constrained by the HSC-Y3, Planck, ACT DR6, and DESI data. The reported Delta chi^2 values and the approximate 2 sigma preference for beta > 0 are statements about the fitted posterior, so they are not 'predictions' that reduce by construction to an input. Equation (7) relating the gamma and DETG models is derived, not assumed, and it is used only for interpretation. The fixed nuisance parameters taken from the LambdaCDM MAP in Li et al. (2023) are a modeling choice that could affect the statistical conclusions, but this is a robustness concern rather than circularity: the paper does not define its target result in terms of those nuisance values, and the extended-model parameters are still varied in the fit. Self-citations to Terasawa et al. and Li et al. are used for data vectors, scale cuts, and baryonic modeling conventions, not as load-bearing theorems. No equation, prior, or cited result in the paper makes the central claim equivalent to an input by definition.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The central comparison rests on the DETG ansatz, with beta fitted to the data and p chosen by hand. The model assumes a fixed LambdaCDM background and relies on the HMCODE nonlinear mapping. No new particles, forces, or entities are introduced.

free parameters (6)
  • DETG beta = 0.443 (mode) for p=5; 0.578 for p=6
    Amplitude of the late-time suppression of the linear power spectrum; fitted to the combined dataset; the posterior mode beta > 0 drives the main claim.
  • DETG p
    Discrete index controlling the redshift evolution of the suppression; chosen from p=1..6 and highlighted for p=5; not sampled with a prior, so the multiple-choice interpretation is not corrected.
  • gamma growth index = 0.690 (+0.133/-0.109)
    Growth index in the alternative gamma model; fitted and found above the GR value 0.55, but with weak evidence.
  • A_b (HMCODE16) = 1.03 (best-fit)
    Amplitude of baryonic feedback in the HMCODE16 model; fitted in the baryonic-extension comparison.
  • M_nu = 0.0026 eV (best-fit)
    Total neutrino mass; fitted in the varying-neutrino model, found consistent with zero and disfavored.
  • HMCODE20 six baryonic parameters (B0, Bz, f*,0, f*,z, log10(Mb,0), Mb,z)
    Flexible baryonic feedback parameters; fitted in one comparison model; modes not reported in Table II.
assumptions (4)
  • ad hoc to paper The DETG ansatz alpha(z) = 1 - beta (Omega_DE(z)/Omega_DE(0))^p captures modified gravity effects on the linear power spectrum.
    Phenomenological parameterization borrowed from Lin et al. (2024), not derived from a theory; the paper's central comparison assumes this form.
  • domain assumption Background expansion is fixed to flat LambdaCDM even when structure growth is modified.
    Section II.B states that Omega_DE(z) is given by the background LambdaCDM model; modified gravity models that also alter the expansion history are excluded.
  • domain assumption The nonlinear matter power spectrum is computed from the modified linear spectrum using the HMCODE16 mapping.
    Assumes the halo-model mapping calibrated for LambdaCDM remains valid for modified growth models and for the DETG power suppression.
  • domain assumption CMB primary spectra remain essentially unchanged under modified growth; only the growth factor affects late-time lensing and shear.
    Section II.B.1 normalizes D(a) so that D/a is constant in the matter-dominated regime, adopting the CMB normalization for the linear power spectrum.

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Pith. "Pith review of Late-time suppression of structure growth as a solution for the $S_8$ tension." pith.science (2026). https://pith.science/paper/ZTPFPT6C

@misc{pith2026250509176,
  author       = {Pith},
  title        = {Pith review of: Late-time suppression of structure growth as a solution for the $S_8$ tension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZTPFPT6C}},
  note         = {Machine review of arXiv:2505.09176}
}
abstract

The $S_8$ value inferred from the Subaru Hyper Suprime-Cam (HSC) Year 3 cosmic shear data, under the assumption of the flat $\Lambda$CDM model, is 2-3$\sigma$ lower than that inferred from observations of the early-time universe, such as cosmic microwave background (CMB) anisotropy data. Resolving the $S_8$ tension requires a scenario in which structure formation on small scales is suppressed in the late universe. As potential solutions, we consider extended models both within and beyond the $\Lambda$CDM model -- models that incorporate parameterized baryonic feedback effects, the effect of varying neutrino mass, and modified structure growth, each of which can lead to a suppression of structure growth at lower redshifts, with its own distinct scale- and redshift-dependencies. In particular, we consider phenomenological modified gravity models in which the suppression of structure growth is triggered at lower redshifts, as dark energy ($\Lambda$) begins to dominate the background expansion. We show that the modified growth factor models -- especially those featuring more rapid growth suppression at lower redshifts -- provide an improved fit to the combined datasets of the HSC-Y3 cosmic shear correlation functions, the Planck CMB, and the ACT DR6 CMB lensing, compared to the fiducial $\Lambda$CDM model and the models including the baryonic effects or the massive neutrino effect within the the $\Lambda$CDM framework.

Figures

Figures reproduced from arXiv: 2505.09176 by the authors.

Figure 1
Figure 1. FIG. 1. Fractional change in the nonlinear matter power spectrum [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Fractional change in the linear matter power spectrum [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Fractional change in the cosmic shear two-point correlation functions (2PCFs), [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The 1D and 2D posteriors obtained from the parameter in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The 1D and 2D posteriors obtained from the parameter in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The 1D and 2D posteriors obtained from the parameter inference of the [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

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