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The paper claims that the growth of cosmic structures is mildly scale-dependent, finding a 2.2σ preference for a non-zero scale-dependent correction to the growth equation when combining fσ8, cosmic chronometer, supernova, and CMB data, whi

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2026-08-04 12:53 UTC pith:77OT44B5

load-bearing objection Read this as a constraints paper, not as evidence for new physics: the first joint CC+SN+fσ8+CMB constraint on the Dent et al. scale-dependent amplitude is useful and honestly analyzed, but the 2.2σ hint rests on a fixed, untested k choice and the paper's own Bayes factor is inconclusive. the 4 major comments →

arxiv 2608.02175 v1 pith:77OT44B5 submitted 2026-08-03 astro-ph.CO

Testing Scale-Dependent Suppression of Structure Growth in the Linear Regime

Fernanda Oliveira , Miguel A. Sabogal , Felipe Avila , Rafael C. Nunes , Armando Bernui This is my paper
classification astro-ph.CO
keywords cosmologylarge-scale structurescale-dependent growthmodified gravityfσ8structure growth suppressionMCMCΛCDM
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tests whether the apparent suppression in the growth rate of cosmic structures, seen in fσ8 data, can be explained by a scale-dependent modification of the growth equation. It introduces an effective gravitational coupling that weakens gravity on large scales and fits the model to a combination of growth-rate, expansion-history, and CMB data. The central result is a 2.2σ indication for a non-zero scale-dependent amplitude when CMB data are included, implying suppressed clustering at large scales and earlier times. Without CMB, the data give only upper limits, and the standard ΛCDM model remains fully consistent; the scale-dependent model fits the growth data somewhat better but Bayesian evidence is inconclusive. A sympathetic reader would see this as a quantitative, cautious hint that linear structure growth may depart from strict scale independence, worth testing with future surveys.

Core claim

The paper claims that linear matter clustering at scales near k=0.1 h/Mpc is plausibly suppressed relative to the GR/ΛCDM prediction, with an effective gravitational coupling Geff=G/(1+Aξ) where ξ∝(1+z)/k². Using a joint MCMC analysis of 35 uncorrelated fσ8 measurements, cosmic chronometer H(z) data, Pantheon+ supernovae, and CMB power spectra and lensing, the inferred amplitude A=4425±2000 (equivalently B=0.36+0.14−0.12) gives a 2.2σ preference for a non-zero scale-dependent term. The suppression is stronger during the matter-dominated era than in the dark-energy-dominated epoch, and the derived S8=0.831±0.011 matches the CMB-inferred value without exacerbating either the S8 or H0 tensions.

What carries the argument

The central object is a scale-dependent effective gravitational coupling, Geff(t,k)=G/(1+ξ(t,k)), inserted into the linear growth equation for matter density perturbations. The correction is ξ=3H0²Ωm0/(a c² k²), so it grows as (1+z)/k², vanishing at small scales and low redshifts. The analysis fixes k=0.1 h/Mpc and samples an amplitude parameter B=ln(1+A/10⁴), which enforces A≥0 and reduces to ΛCDM at B=0. This machinery converts a growth-rate anomaly into a constraint on large-scale gravitational weakening, and the k-dependence is what makes the model distinguishable from standard growth.

Load-bearing premise

The analysis fixes the comoving wavenumber to k=0.1 h/Mpc for all fσ8 measurements and asserts, without showing a scan, that other reasonable choices of k have no appreciable effect—yet since the correction ξ is proportional to 1/k², the inferred amplitude A and the 2.2σ evidence are tied to that assumed scale.

What would settle it

Repeat the MCMC analysis at k=0.05 and k=0.2 h/Mpc (a k-scan) and check whether the posterior for A or B shifts by more than the reported uncertainties, or whether the 2.2σ preference for A>0 disappears; a strong k-dependence of the inferred amplitude would show the result is not robust to the fixed scale. A direct test would be measuring fσ8 in two or more independent k-bins within the same redshift range to look for the predicted (1+z)/k² suppression pattern.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the 2.2σ signal is real, linear matter clustering at large scales (k≈0.1 h/Mpc) is suppressed relative to GR/ΛCDM, equivalent to an effective gravitational coupling below G at early times and large scales.
  • The suppression is stronger in the matter-dominated era than in the dark-energy-dominated epoch, so high-redshift growth measurements are the most sensitive probes of the effect.
  • The model keeps S8 and H0 consistent with CMB-inferred values, meaning a scale-dependent growth term does not worsen the current cosmological tensions.
  • The non-CMB dataset combinations yield upper limits on A (A≲6700–9900 at 95% CL), providing a quantitative bound on possible large-scale deviations at the O(10%) level in clustering strength.
  • Future surveys probing wider k-ranges and higher redshifts should distinguish the scale-dependent growth model from ΛCDM more decisively than current data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The 2.2σ evidence appears only when CMB data are included; without it, the same parametrization yields only upper limits. This suggests the signal is driven by the CMB's tight anchoring of Ωm,0 and σ8,0, and could be sensitive to CMB calibration or systematics rather than a genuine scale-dependent growth effect.
  • Because the correction scales as 1/k², the inferred amplitude A is inversely tied to the assumed wavenumber k=0.1 h/Mpc. A natural extension is to treat k as a free parameter or fit the growth rate in several k-bins; if A shifts with k as 1/k², the current parametrization may be absorbing a scale-mismatch rather than a physical effect.
  • A scale-dependent weakening of gravity on large scales would also affect weak lensing shear, CMB lensing, and void statistics on comparable scales; cross-correlating these probes with fσ8 in matched k-bins would provide an independent test of the claim.
  • The inconclusive Bayes factor (|lnB|<1) suggests the extra parameter is not yet justified on model-selection grounds; future redshift-space distortion surveys with finer k-resolution and lower systematics are needed to push the evidence beyond the current threshold.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a phenomenological scale-dependent modification of the linear growth of matter perturbations, replacing the standard gravitational coupling G with G_eff(t,k)=G/[1+A ξ(t,k)], where ξ=3 H_0^2 Ω_m0/(a c^2 k^2). Using CLASS and MontePython, the authors jointly fit this model to 35 uncorrelated fσ8(z) measurements, cosmic chronometer H(z) data, PantheonPlus supernovae, and Planck 2018 CMB data. In the full dataset combination they report A=4425±2000 (B=0.36^{+0.14}_{-0.12}), interpreted as a 2.2σ preference for scale-dependent suppression of growth, while also reporting that the Bayesian evidence is inconclusive (ln B_ij=-0.87) and that the model does not alleviate the S8 or H0 tensions. The paper emphasizes that the suppression is stronger at z≳0.5 and that the model remains consistent with Planck values of S8 and H0.

Significance. If the central 2.2σ result were robust, it would be an interesting hint of scale-dependent gravitational behavior on linear scales, with implications for modified gravity and for interpretations of fσ8 measurements. The paper has several strengths: it uses public, well-established Boltzmann and MCMC tools; it gives a clear χ² breakdown per dataset; and it honestly reports the inconclusive Bayes factor, which tempers the headline significance. However, the central claim currently rests on a fixed, unvalidated choice of k, a one-sided prior on the amplitude, and a non-independent comparison with Planck-derived S8 and H0. With the evidence as presented, the paper is more a proof-of-concept constraint than a robust detection. The question of whether the claimed 2.2σ is scale-choice dependent is directly testable and should be resolved before the result is used as evidence for new physics.

major comments (4)
  1. [Section 2, Eq. (5) and text after Eq. (12)] The fixed choice k=0.1 h/Mpc is load-bearing for the central result. Because ξ∝1/k², the amplitude A and the posterior for B are not invariant under a change of k: the same physical suppression at k=0.05 h/Mpc corresponds to A~1100 and B~0.10, not B=0.36. The sentence 'Other reasonable choices of k have no appreciable effect on our results' is not a substitute for a k-scan. Please show the marginalized posterior of A (or B) and the resulting significance for at least k=0.05, 0.07, 0.1, 0.15, 0.2 h/Mpc, and state whether the 2.2σ indication persists across that range.
  2. [Eq. (9) and Table 3] The 2.2σ is a one-sided posterior probability for A>0 from a flat prior on B∈[0,3], not a model-comparison significance. The paper's own Bayes factor, ln B_ij=-0.87, is described as inconclusive under Kass & Raftery. This internal tension should be addressed directly: either the headline should be framed as 'a mild one-sided preference within a fixed parametrization' or the paper should justify why the Bayesian evidence is being discounted. Reporting a two-sided posterior probability or ΔBIC/ΔAIC would clarify what the data actually support.
  3. [Section 3, S8 and H0 consistency discussion] The agreement of S8=0.831±0.011 and H0=67.38±0.52 with Planck values is presented as validation, but Planck CMB data are included in the same joint fit. This agreement is therefore not independent and cannot be used to argue that the model preserves the Planck picture. Please rephrase this as internal consistency of the joint fit, or perform a fit excluding CMB data if the goal is an independent cross-check.
  4. [Section 2.1 and Figure 2] The model predicts fσ8(k,z), but the 35 fσ8 measurements are reported as single values per redshift, each averaged over a survey-dependent k-window. Comparing the model at k=0.1 h/Mpc to all points assumes that every RSD measurement is effectively at that single wavenumber. This assumption is not tested. The paper should at least discuss the typical effective k of the surveys in the compilation and ideally marginalize over or propagate the k-window uncertainty.
minor comments (5)
  1. [Abstract] The abstract says 'evidence for scale-dependent growth at 2.2σ' while Section 3 repeatedly uses 'indication' and 'hint'. Please harmonize the language to match the statistical strength and the inconclusive Bayes factor.
  2. [Eq. (9)] The footnote explaining the 10^4 normalization says ξ is 'typically of order 10^-5', but the units in Eq. (5) are not explicitly stated. Please state that k is in h/Mpc and that ξ is dimensionless with the conventions used, to avoid confusion.
  3. [Section 3, Table 3] The Δχ²_min values are useful, but the total Δχ²_min=-6.04 is compared with a Bayesian ln B_ij=-0.87 without giving the number of degrees of freedom or the effective penalty. A brief explanation of why the χ² improvement does not translate into positive evidence would help the reader.
  4. [Figure 1] The caption says 'dark blue curves' for fσ8 only, but the figure legend uses multiple colors. Please ensure the color description matches the actual plot, and consider labeling curves directly for accessibility.
  5. [References] Some references are incomplete or inconsistently formatted (e.g., 'Dent et al. 2009' is cited with arXiv numbers but some entries lack page numbers or journal identifiers). A final reference cleanup is needed.

Circularity Check

1 steps flagged

Core inference is a joint fit, not a prediction; only the Planck-consistency 'validation' is co-fit and partly circular.

specific steps
  1. fitted input called prediction [Section 3, after Eq. (13)]
    "From our MCMC analysis, we obtain S 8 =0.831±0.011, which is in excellent agreement with the value reported by the Planck collaboration, S 8 =0.832±0.013 (Planck Collaboration et al., 2020). Similarly, the inferred Hubble constant, H 0 =67.38±0.52 km s −1 Mpc−1, is fully consistent with the Planck result, H 0 =67.36±0.54 km s−1 Mpc−1 (Planck Collaboration et al., 2020). These agreements indicate that the proposed scale-dependent extension preserves the successful description of early- and late-time background observables."

    S8 and H0 are derived from the same MCMC whose likelihood includes the Planck 2018 TT/TE/EE and lensing spectra. Because the background is fixed to ΛCDM and the parameter B does not alter H(a), these quantities are essentially read off from the Planck data inside the fit. Presenting the resulting agreement with Planck as a validation of the model is therefore a co-fit consistency check, not an independent test: the model is not predicting these numbers, it is returning the fitted input. This is ancillary to the A>0 result and does not by itself invalidate the parameter estimate, but it is a circular form of validation.

full rationale

The central claim is a posterior statement: after jointly fitting B (or A) to fσ8+CC+SN+CMB, the authors find B=0.36+0.14−0.12, a 2.2σ indication that A>0. This is a parameter estimate from a fit, not an out-of-sample prediction, so it is not circular in the strict derivation-chain sense—the fσ8 data alone yield only upper limits on A, showing the detection is not hard-wired into the model. The scale-dependent ξ(t,k) is explicitly adopted from Dent et al. (2009) as a parametrization, not derived from first principles, and the paper is candid about the inconclusive Bayes factor (ln B_ij = −0.87). No load-bearing self-citation appears: the fσ8 compilation (Skara & Perivolaropoulos 2020) and the growth-equation ansatz (Dent et al. 2009) are external, while the many Oliveira/Sabogal/Nunes/Bernui citations are contextual rather than load-bearing. The fixed-k=0.1 h/Mpc choice is asserted without a k-scan ('Other reasonable choices of k have no appreciable effect on our results'), and since ξ∝1/k^2 the quoted A is k-specific; however, this is a robustness/prior-dependence caveat, not a circular reduction. The only identifiable circular element is the presentation of Planck consistency as validation while Planck data are inside the same joint fit; this is minor and does not drive the A>0 evidence, hence the low score.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 1 invented entities

The central result is a fit of one new amplitude A in a known parametrization, on top of the standard ΛCDM parameters. The only genuinely new free parameter is A; k is fixed by hand. No new entities beyond an effective Geff function are introduced, and that function has no independent falsifiable handle outside the fitted data.

free parameters (3)
  • A (scale-dependent amplitude) = A=4425±2000 with CMB; A<6703-9937 (95% CL) without CMB
    Central new parameter; promotes ξ in Eq. (5); prior on B=ln(1+A/10^4) uniform over [0,3] enforces A≥0.
  • k (comoving wavenumber) = fixed to 0.1 h/Mpc
    Chosen by hand; all fσ8 points evaluated at this single scale; no k-scan is shown despite the paper's claim that other choices do not matter.
  • ΛCDM baseline parameters (ωb, ωcdm, 100θs, τreio, As, ns, Ωm0, σ8,0, H0, MB) = Ωm0=0.3152±0.0072, σ8=0.8111±0.0059, H0=67.38±0.52, MB=-19.439±0.015; see Table 2
    Standard cosmological and nuisance parameters sampled jointly; needed to evaluate the growth and likelihoods, and they anchor the constraints on A.
axioms (5)
  • domain assumption Linear perturbation growth equation (Eq. 2) is valid in the sub-horizon, quasi-static approximation.
    Section 2, Eq. (2); standard but breaks near horizon scales, which motivates the added ξ correction.
  • domain assumption The functional form ξ(t,k)=3H0²Ωm0/(a c² k²) correctly parametrizes scale-dependent growth (Dent et al. 2009).
    Section 2, Eq. (5); adopted from prior literature, not derived from an action; the paper's central result depends entirely on this form.
  • domain assumption Modified growth can be implemented in CLASS with a GR background while leaving CMB/lensing predictions otherwise standard.
    Section 2; no consistency check of the Poisson/GR relation for the ad hoc Geff modification is provided.
  • domain assumption The fσ8 compilation of Skara & Perivolaropoulos (2020) is uncorrelated and free of unmodeled systematics.
    Section 2.1; the 2.2σ signal is driven by this dataset, so correlated errors or systematics would change the significance.
  • standard math Spatially flat ΛCDM background, H(a)=H0√(Ωm0 a^-3 + (1-Ωm0)), is assumed.
    Section 2; standard background evolution, long after radiation domination.
invented entities (1)
  • Effective gravitational coupling Geff(t,k)=G/(1+Aξ(t,k)) no independent evidence
    purpose: Phenomenologically suppress gravitational clustering at large scales and early times.
    Introduced via Eqs. (4)-(8) as an ad hoc modification; no independent prediction outside the fitted growth data, and no new particle or force beyond the parametrization.

pith-pipeline@v1.3.0-daily-deepseek · 17033 in / 15752 out tokens · 120795 ms · 2026-08-04T12:53:01.787224+00:00 · methodology

0 comments
read the original abstract

We investigate recent reports of a suppression in the growth rate of cosmic structures inferred from analyses of the $[f\sigma_8](z)$ dataset. To address this issue, we explore the hypothesis that the evolution of matter clustering is more accurately described within the framework of scale-dependent modified gravity. We perform a joint analysis of $[f\sigma_8](z)$, cosmic chronometer $H(z)$ measurements, luminosity distance data, and CMB observations using Markov Chain Monte Carlo techniques to constrain the parameters of a scale-dependent cosmological model and investigate its impact on the evolution of $[f\sigma_8](z)$. Our results indicate that the suppression of the growth rate of large-scale structures is more pronounced during the matter-dominated era than in the dark-energy-dominated epoch. We find evidence for scale-dependent growth at a statistical significance of $2.2\, \sigma$. In addition, we constrain the $S_8$ parameter and find it to be consistent with the value inferred from the CMB observations of the Planck Collaboration. Overall, our analysis shows that $k$-dependent growth models provide a viable explanation for the observed clustering of matter without exacerbating the current cosmological tensions.

Figures

Figures reproduced from arXiv: 2608.02175 by Armando Bernui, Felipe Avila, Fernanda Oliveira, Miguel A. Sabogal, Rafael C. Nunes.

Figure 1
Figure 1. Figure 1: Marginalized posterior distributions for a subset of the parameters investigated in this work. The dark blue curves correspond to the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison between [fσ8](z) prediction using fσ8+CC+SN+CMB data for the scale-dependent model and the ΛCDM model. The shaded areas represent the 2σ CL region. The bottom panel shows the relative difference between the data points and the two models analyzed. In [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

discussion (0)

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Reference graph

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