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arxiv: 2605.13914 · v1 · submitted 2026-05-13 · 🌌 astro-ph.CO · gr-qc· hep-ph· hep-th

Recognition: 2 theorem links

· Lean Theorem

The Amplitude-Growth Degeneracy and Implied A_s Diagnostic for Background-Inert Modified Gravity

Ameya Kolhatkar , P. K. Sahoo
Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:44 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-phhep-th
keywords f(Q) gravitybackground-inert couplingamplitude-growth degeneracyAs diagnosticσ80growth factor D0CMB distance priors
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The pith

Background-inert f(Q) gravity couplings degenerate with σ80 through a deeper As-D0(λ) link when CMB distance priors are used.

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

The paper establishes that any background-inert perturbative coupling λ in coincident f(Q) gravity produces a degeneracy with the clustering amplitude σ80 when analyses rely on compressed CMB distance priors. This surface degeneracy arises directly from a more fundamental relation between the primordial amplitude As and the present-day growth factor D0(λ). The authors supply a consistency diagnostic that computes the As value required to reproduce a given sampled σ80 and apply it to ΛCDM and hybrid models. They show that the λ0√(QQ0) term drives σ80 to unphysical levels and forces As upward by 20-30 percent, creating tension with Planck; imposing ln(As) priors restores baseline parameters while penalizing the extra parameter by roughly two units.

Core claim

Any background-inert λ in coincident f(Q) gravity exhibits a degeneracy with σ80 under compressed CMB distance priors; this is the direct manifestation of an As-D0(λ) degeneracy. The consistency check computes the As needed to match the sampler’s σ80 prediction. Inclusion of the λ0√(QQ0) correction inflates σ80 and raises As by 20-30 percent into 1.7-2.2σ tension with Planck. Fixing ln(As) to its Planck 1σ range returns all parameters to baseline values, with each extended model penalized by about two units per extra parameter.

What carries the argument

The As-D0(λ) degeneracy, which links the primordial power-spectrum amplitude directly to the present-day growth factor produced by a background-inert perturbative coupling.

If this is right

  • Adding the λ0√(QQ0) term inflates σ80 to unphysical values unless As is increased by 20-30 percent.
  • The increase in As places the models in 1.7-2.2σ tension with Planck constraints.
  • Imposing Planck 1σ priors on ln(As) returns all other parameters to their baseline values.
  • Each extended model is penalized by roughly two units per extra parameter in model-selection statistics.
  • The ΛCDM+λ0+ln(As) combination with SDSS DR16 data shows a weak preference over vanilla ΛCDM.

Where Pith is reading between the lines

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

  • The same As-D0(λ) degeneracy is likely to appear in any modified-gravity model whose linear growth is altered while the background expansion remains unchanged.
  • The proposed As diagnostic offers a quick consistency test that can be applied to growth-rate data sets before running expensive full likelihood chains.
  • Future surveys that tighten both σ8 and As measurements independently could break or confirm the degeneracy without relying on compressed CMB priors.
  • Analyses that combine growth data with distance priors should routinely check whether an inferred rise in σ80 is actually an artifact of an unadjusted As.

Load-bearing premise

The perturbative coupling stays strictly background-inert and the compressed CMB distance priors contain enough information to reveal the degeneracy without a full likelihood analysis.

What would settle it

A full MCMC run with complete Planck likelihoods (instead of compressed distance priors) that shows no upward shift in As and no unphysical inflation in σ80 when the λ0√(QQ0) term is added.

Figures

Figures reproduced from arXiv: 2605.13914 by Ameya Kolhatkar, P. K. Sahoo.

Figure 1
Figure 1. Figure 1: Posteriors for BD2R and BSDp combinations 1 0 1 2 3 0.5 0.6 0.7 0.8 0.9 1.0 G(z) f(Q) f(Q) + 0 f(Q) + 0 + ln(As) CDM CDM+ 0 CDM+ 0 + ln(As) BD2R BSDp [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The reconstructed µG(z) plot for all the model and dataset combinations [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Overlay plot for fσ8(z) for all the model and dataset combinations 0.0 0.5 1.0 1.5 2.0 2.5 Implied As [×10 9 ] CDM CDM + 0 CDM + 0 + ln(As) f(Q) f(Q) + 0 f(Q) + 0 + ln(As) 2.101 2.706 2.108 2.099 2.749 2.108 Planck 2018 Planck 1 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The implied As for BD2R 0.0 0.5 1.0 1.5 2.0 2.5 Implied As [×10 9 ] CDM CDM + 0 CDM + 0 + ln(As) f(Q) f(Q) + 0 f(Q) + 0 + ln(As) 2.101 2.551 2.110 2.113 2.609 2.111 Planck 2018 Planck 1 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The implied As for BSDp 80 100 120 140 160 180 H(z) DDTB+RSD CDM CDM + 0 f(Q) f(Q) + 0 CDM+ 0 + ln(As) f(Q) + 0 + ln(As) 1.5 0.0 1.5 H/Href [%] 80 100 120 140 160 180 H(z) SDSS BAO+ 0.0 0.5 1.0 1.5 Redshift z 2 0 2 H/Href [%] [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Overlay plot for the distance modulus µ(z) for all the model and dataset combinations and DIC are penalized by 2 units per free parameter and show moderate preference for the λ0 variants. The BIC imposes a heavier penalty of ln(N), showing a substan￾tially weaker preference. Bayesian evidence log Z ∼ 3−5 corresponds to a moderate preference for λ0 variants. Even though all four metrics evaluate the same de… view at source ↗
Figure 8
Figure 8. Figure 8: Overlay plot for q(z) for all the model and dataset combinations this model even after the degeneracy is broken. Since the ln(As) prior prevents unphysical statistics from driving the parameters, this preference does not originate from amplitude inflation. The structural difference between the two pipelines – BD2R with isolated data points for BAO and RSD and BSDp with covariant BAO+RSD measurements – may … view at source ↗
Figure 10
Figure 10. Figure 10: Whisker plot for σ80 for all the model and dataset combinations pling solely altering the effective gravitational coupling µG(z, λ) = Gef f (z, λ)/GN , and completely invisible to the background expansion. Both Li et al. (2025) and Kolhatkar & Sahoo (2026) investigate the √ Q pertur￾bation correction using RSD data and find a mild am￾plification of σ80 in each case; Li et al. (2025) explicitly notes a res… view at source ↗
Figure 9
Figure 9. Figure 9: Overlay plot for ωeff (z) for all the model and dataset combinations tilts only slightly in the weak side. Independent vali￾dation using the full-shape CMB likelihood and Stage IV growth data is required before this result can be re￾garded as a physical signal. 5.4. Scope and Applicability of the diagnostic test The conditions stated in Subsection 2.3 are sufficient (not necessary) for the application of t… view at source ↗
Figure 11
Figure 11. Figure 11: Pull plot for BD2R the compressed CMB priors only constrain background geometry through the shift parameters R and la, the pri￾mordial amplitude is implicitly fixed to AP lanck s , driving the sampler into exploiting the σ80 − λ0 degeneracy val￾ley. The effective gravitational coupling is suppressed at late-times and the sampler compensates by inflating σ80 while simultaneously preferring larger λ0 values… view at source ↗
Figure 12
Figure 12. Figure 12: Pull plot for BSDp provement costs a 20% − 30% higher amplitude as com￾pared to AP lanck s at a 1.7σ − 2.2σ tension depending on the dataset and model combination. The evidence metrics – AIC, BIC, DIC and log Z return moderate to weak preference over the λ0 variants, differentiating the statistics from the physics. There is, however, one exception to this inference – the ΛCDM+λ0 + ln(As) model under BSDp … view at source ↗
read the original abstract

We prove that any background-inert perturbative coupling $ \lambda $ in coincident $ f(Q) $ gravity exhibits a degeneracy with the clustering amplitude $ \sigma_{80} $, when using compressed CMB distance priors. This degeneracy is, in fact, a direct materialization of a more deeper $ A_s-D_0(\lambda) $ degeneracy between the primordial amplitude $ A_s $ and the present day growth factor $ D_0(\lambda) $. We outline a consistency check scheme, applicable to models even outside the $ f(Q) $ class, by computing $ A_s $ needed to reproduce the $ \sigma_{80} $ predicted by the sampler. We perform our analysis with two dataset pipelines, based on the coupled/decoupled $ f\sigma_8(z) $ data. To ensure theoretical diversity, we include $ \Lambda $CDM and the Hybrid model in the $ f(Q) $ framework. Our results illustrate that adding the $ \lambda_0\sqrt{QQ_0} $ correction to the models inflates $ \sigma_{80} $ to unphysical values, while showing moderate evidence in favor of the said models. However, this results in an increase of $ 20\%-30\% $ in $ A_s $ in $ 1.7\sigma-2.2\sigma $ tension with Planck values. We utilize the $ 1\sigma $ $ \ln(A_s) $ constraints from Planck as priors in order to fix the artificial increase in $ \sigma_{80} $ and find that all the constrained parameters return to their baseline values. Each model is penalized by around $ 2 $ units per extra parameter. Interestingly, the $ \Lambda $CDM$ +\lambda_0+\ln(A_s) $ + SDSS DR16 combination shows a weak preference over the vanilla $ \Lambda $CDM model, validated by the values of $ \log\mathcal{Z},\;AIC,\;DIC, $ and BIC.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proves that any background-inert perturbative coupling λ in coincident f(Q) gravity exhibits a degeneracy with the clustering amplitude σ80 when using compressed CMB distance priors; this is presented as a direct consequence of a deeper As-D0(λ) degeneracy. It introduces a consistency-check scheme that recomputes the As value needed to reproduce the sampler's σ80 prediction, applies the analysis to ΛCDM and a Hybrid f(Q) model using two fσ8 pipelines (coupled/decoupled), reports that the λ0√(QQ0) correction inflates σ80 to unphysical values while producing 20-30% As shifts in 1.7-2.2σ tension with Planck, and shows that imposing 1σ ln(As) priors restores baseline parameter values. Model-selection metrics (logZ, AIC, DIC, BIC) penalize each model by ~2 units per extra parameter, with a weak preference noted for one ΛCDM+λ0+ln(As)+SDSS DR16 combination.

Significance. If the degeneracy proof and consistency check hold under the stated priors, the work supplies a practical diagnostic for As-related degeneracies in background-inert modified-gravity models and a reusable scheme applicable beyond the f(Q) class. The explicit demonstration that ln(As) priors can restore baseline σ80 values, together with the quantitative model-selection penalties, provides falsifiable guidance for future analyses. The limitation to compressed CMB priors, however, restricts the generality of the central claim.

major comments (3)
  1. [Degeneracy proof] The proof that the λ-σ80 degeneracy is a direct materialization of the As-D0(λ) degeneracy (abstract and §3) is stated but not accompanied by the explicit algebraic steps linking D0(λ) to the growth equation under strictly background-inert conditions; without these steps the claim cannot be verified from the given information.
  2. [Consistency check] The consistency check recomputes As to match the sampled σ80 (abstract and results section); this procedure must be shown to be independent of the original fit rather than tautological, especially given the post-hoc imposition of ln(As) priors that restores baseline values.
  3. [CMB priors and data pipelines] All numerical results (σ80 inflation, 20-30% As shift, model-selection outcomes) are obtained exclusively with compressed CMB distance priors; the manuscript does not test or discuss whether the full Planck likelihood supplies additional growth or correlation information that would lift the degeneracy.
minor comments (2)
  1. The abstract contains the phrasing 'a more deeper'; this should be corrected to 'a deeper'.
  2. [Model selection] The exact numerical values of logZ, AIC, DIC and BIC for each model combination are summarized but not tabulated; a table would improve clarity and allow direct comparison of the ~2-unit penalty per extra parameter.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough and insightful report. We address each of the major comments point by point below, providing clarifications and indicating where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: [Degeneracy proof] The proof that the λ-σ80 degeneracy is a direct materialization of the As-D0(λ) degeneracy (abstract and §3) is stated but not accompanied by the explicit algebraic steps linking D0(λ) to the growth equation under strictly background-inert conditions; without these steps the claim cannot be verified from the given information.

    Authors: We acknowledge that the explicit algebraic steps were not provided in the original manuscript. In the revised version, we will expand §3 to include the detailed derivation. Starting from the linear growth equation in the background-inert coincident f(Q) framework, we will show how the present-day growth factor D0(λ) modifies the normalization of σ80, leading directly to the degeneracy with As when using compressed CMB priors. This will allow verification of the claim. revision: yes

  2. Referee: [Consistency check] The consistency check recomputes As to match the sampled σ80 (abstract and results section); this procedure must be shown to be independent of the original fit rather than tautological, especially given the post-hoc imposition of ln(As) priors that restores baseline values.

    Authors: The consistency check is designed as an independent diagnostic tool. After performing the MCMC fit allowing both As and σ80 to vary freely, we use the sampled σ80 value to back-calculate the As that would be required to match it given the model's D0(λ). This is not tautological because it highlights the implied shift in As due to the degeneracy. The subsequent imposition of ln(As) priors is a separate analysis to test the impact on parameter recovery. We will add explicit equations and a description in the revised manuscript to clarify this independence and avoid any perception of circularity. revision: yes

  3. Referee: [CMB priors and data pipelines] All numerical results (σ80 inflation, 20-30% As shift, model-selection outcomes) are obtained exclusively with compressed CMB distance priors; the manuscript does not test or discuss whether the full Planck likelihood supplies additional growth or correlation information that would lift the degeneracy.

    Authors: This is a valid point regarding the scope of our analysis. We chose compressed CMB distance priors to focus on the background-inert effects in a controlled setting, consistent with many prior studies on modified gravity. We will revise the manuscript to include a discussion of this limitation, noting that the full Planck likelihood may include additional constraints from growth and cross-correlations that could potentially break the degeneracy. We suggest this as an avenue for future investigation but do not perform the full likelihood analysis here, as it would require significant additional computational resources. revision: partial

Circularity Check

0 steps flagged

No significant circularity: degeneracy shown via MCMC under compressed priors; As diagnostic is post-hoc and independent

full rationale

The central claim is established by running MCMC samplers on two fσ8 pipelines with compressed CMB distance priors, demonstrating that background-inert λ inflates σ80 and requires a compensating 20-30% shift in As to match the sampled growth. The consistency check recomputes the As value needed to reproduce the sampler's σ80 prediction using the growth factor D0(λ); this is presented as a diagnostic applicable beyond f(Q) and does not reduce the sampled posterior to its inputs by construction. No equations equate the prediction to the fit parameters tautologically, no self-citations carry the uniqueness or ansatz load, and the numerical results (inflation of σ80, return to baseline under ln(As) prior, information criteria) are obtained from standard sampling rather than renaming or self-definition. The derivation remains self-contained against the stated priors and datasets.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the background-inert assumption for λ and the sufficiency of compressed CMB priors; λ0 is introduced as an additional free parameter whose value is constrained by the data.

free parameters (1)
  • λ0
    Perturbative coupling strength in the λ0 √(Q Q0) correction term, introduced to modify the f(Q) action.
axioms (2)
  • domain assumption The perturbative coupling λ is background-inert
    Invoked to ensure the degeneracy arises purely from the growth sector without altering background expansion.
  • domain assumption Compressed CMB distance priors suffice to reveal the degeneracy
    Used as the primary CMB constraint in the fitting pipelines.

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Reference graph

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