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REVIEW 3 major objections 7 minor 61 references

Scale-dependent and background-preserving gravity from an action: cosmological tests

T0 review · 3 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper tests a gravity model whose constants G and Λ run with scale, and finds the single free parameter is pinned near one part in 100,000, leaving the Hubble and σ8 tensions unresolved.

desk verdict A well-executed null result for one specific RGGR model, but the paper's |ν|≲10^-5 headline is contradicted by its own Table 1 (the 2σ bound is ~2×10^-4); fix that and it earns referee time. read the letter →

arxiv 2411.12097 v2 pith:YYPIJ6T2 submitted 2024-11-18 gr-qc

Wiliam S. Hipólito-Ricaldi , Rodrigo von Marttens , Felipe de Melo-Santos , Davi C. Rodrigues This is my paper · ORCID
classification gr-qc
keywords renormalizationgroupimprovedgravityscale-dependentgravitationalconstantrunningcosmologicalperturbationsCMBacousticoscillationsHubbletensionsigma8MarkovChainMonteCarlo
topics Dark MatterThe Hubble Tension
open problems Dark Matter
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

Quantum field theory suggests that physical constants run with energy scale, and extending that idea to gravity at cosmic scales motivates the model studied here: the gravitational constant $G$ and the cosmological constant $\Lambda$ vary with scale at the level of perturbations, while the background expansion stays exactly general relativity. The paper builds this running into an action whose scale definitions are enforced by Lagrange multipliers, and derives the full perturbative equations; a single dimensionless parameter $\nu$ carries every deviation from general relativity. Fitting Planck 2018 CMB, BAO, Pantheon supernovae, and redshift-space-distortion data, the joint analysis constrains $\nu = 5.8 \times 10^{-5}$ with a $2\sigma$ uncertainty of $1.7 \times 10^{-4}$, so the general-relativistic value $\nu = 0$ is within $1\sigma$. The CMB acoustic-peak structure does the constraining work: even $|\nu|$ of order $10^{-2}$ visibly shifts the positions and amplitudes of the peaks, which is why CMB data alone already limit $\nu$ to the $10^{-5}$ scale. The paper concludes that, in this two-scale form, the model cannot resolve the Hubble or $\sigma_8$ tensions, although the action framework itself remains theoretically consistent.

What carries the argument

The load-bearing object is the effective action (1) with Lagrange multipliers $\lambda_p$ that enforce the scale-setting conditions $\mu_p = f_p(g,\Psi,\gamma)$ inside the integral, together with the specific two-scale identification: the first scale is set by $W = U^\alpha U^\beta(g_{\alpha\beta} - \gamma_{\alpha\beta}) = -2\psi$ in the Newtonian gauge, and the second scale is tied to the background value of the trace $T$. Expanding $G(\mu_1) = G_0(1 - 2\nu\psi)$ to first order in $W$ with a single constant $\nu$, and fixing $\Lambda$ accordingly, converts the action into a concrete set of $\nu$-corrected perturbation equations: the slip $\phi/\psi = 1 - 2\nu$, a corrected Poisson equation, and a dark-matter continuity equation with source term $-6H\nu\psi$. The analytic solutions $\psi \propto \eta^{\bar\nu}$ with $\bar\nu \approx (6/5)\nu$, and $\delta_c \propto a^{1+\bar\nu/2}$, show how a small $\nu$ amplifies or suppresses potentials and growth. The mechanism that ultimately decides the model's fate is the damped, forced harmonic oscillator for the CMB monopole $\Theta_0$: through $\phi$ and $\psi$, the forcing term acquires $\nu$-dependence that shifts acoustic-peak positions and amplitudes, and Planck-scale data convert that sensitivity into the $10^{-5}$ bound.

What would settle it

Build the same two-scale action with the primary scale tied to the local energy density or the Ricci scalar instead of psi, and fit it to the identical Planck, BAO, Pantheon, and RSD data: a model of that kind that evades the acoustic-peak constraint while shifting H0 by several km/s/Mpc would show that the paper's null result is a property of the scale choice rather than of renormalization-group running itself. Observationally, a CMB experiment with roughly an order-of-magnitude better acoustic-peak calibration than Planck, or growth measurements reaching sub-percent precision in f sigma8, could detect nu at the $10^{-5}$ level; a best-fit value deviating from zero by more than 2 $\sigma$ would overturn the paper's central 'consistent with LambdaCDM' claim.

Watch

Extended reading notes

Core claim

The central claim is a null result expressed as a tight constraint. In the two-scale, action-based RGGR model (renormalization-group improved general relativity), all running is carried by $\nu$, defined through $G_0 G^{-1}(W) = 1 + \nu W$ with $W = -2\psi$, the Newtonian potential perturbation, while $\Lambda$ is tied to the background trace of the energy-momentum tensor. The background equations are identical to general relativity, and only the perturbation sector changes: there is a gravitational slip $\phi/\psi = 1 - 2\nu$, a source term $-6H\nu\psi$ in the cold-dark-matter continuity equation, and a slowly growing potential mode $\psi \propto \eta^{(6/5)\nu}$ that also alters structure growth, $\delta_c \propto a^{1+3\nu/5}$, for positive $\nu$. The joint CMB+BAO+SN+RSD fit returns $\nu = 5.8 \times 10^{-5}$ with $2\sigma$ error $1.7 \times 10^{-4}$, i.e., agreement with $\Lambda$CDM within $1\sigma$; the largest $2\sigma$ shift in $\sigma_8$ is below $0.5\%$ and shifts in $H_0$ are below $1\%$. The paper concludes that, in this implementation, infrared renormalization-group running of $G$ and $\Lambda$ is consistent with the standard model's predictions and does not relieve either of its two main tensions.

Load-bearing premise

The constraint rests on identifying the infrared renormalization-group scale with the Newtonian potential perturbation (W = -2psi) and expanding G and Lambda linearly in that quantity with a single constant nu; if the physical scale were set by a different quantity, or if nu itself ran with scale, the derived perturbation equations and the tight CMB bound would not apply.

Editorial extensions

If this is right

  • The two-scale RGGR model with linear-in-$\psi$ running is observationally indistinguishable from $\Lambda$CDM at current precision, with $\nu$ consistent with zero within $1\sigma$.
  • CMB acoustic-peak structure dominates the constraint; RSD data alone leave $\nu$ almost unconstrained, and earlier hints of a negative $\nu$ from growth data do not survive the joint analysis.
  • The model slightly enlarges the uncertainties on $n_s$, $H_0$, and $\sigma_8$, but every shift relative to $\Lambda$CDM stays below $1\%$, so neither the Hubble nor the $\sigma_8$ tension is alleviated.
  • Any model in this class that modifies the gravitational slip or the dark-matter conservation equation around recombination faces the same tight acoustic-peak bounds.
  • Since the running is forced to be tiny, the framework's phenomenological value would have to come from further generalization, such as different scale settings or additional running scales.

Reading between the lines

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

  • The null result is conditional on the scale choice $W = -2\psi$; a differently scaled variant with the running tied to energy density could in principle move $H_0$ or $\sigma_8$ while leaving the CMB peaks intact, and the same action formalism makes that variant directly testable with the same datasets.
  • The analytic growth law $\delta_c \propto a^{1+3\nu/5}$ implies that even a $\nu$ at the current bound leaves a sub-percent imprint on growth data, so next-generation surveys could tighten the constraint by an order of magnitude or detect a nonzero value.
  • Because the model changes only perturbation-level observables, geometric probes such as BAO peak positions and supernova distances are blind to it; this illustrates that 'background-preserving' modified-gravity models must be tested with clustering and CMB-anisotropy data rather than distance measurements alone.
  • The $\nu$-induced slip between $\phi$ and $\psi$ persists at late times even without anisotropic stress, so a future cosmological measurement of the slip through lensing and dynamics could probe the same parameter from an independent direction.
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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

3 major / 7 minor

Summary. This paper tests a specific action-based implementation of renormalization-group-improved gravity (RGGR) in cosmology. In the model, the gravitational constant G and the cosmological constant Λ depend on two RG scales introduced through Lagrange multipliers; the first scale is set by the Newtonian potential perturbation through W = -2ψ (Eqs. 13-16), so the background cosmology is exactly ΛCDM while the linear perturbations acquire extra terms controlled by a single parameter ν, with ν = 0 recovering GR. The authors derive the modified scalar perturbation equations (25)-(33), give analytic solutions for the matter era (Eqs. 35-45) and for the CMB acoustic monopole (Eqs. 51-53), implement the system in a modified version of CLASS, and run MontePython MCMC against Planck 2018 (TT-TEEE+lensing), Pantheon SNIa, BOSS DR12 BAO, and RSD data. Table 1 reports ν = 8.4×10^-5 ± 1.8×10^-4 (CMB only) and ν = 5.8×10^-5 ± 1.7×10^-4 (joint) at the quoted 2σ level, i.e., consistent with ΛCDM, and the authors conclude that this model cannot resolve the H0 or σ8 tensions and is disfavored relative to ΛCDM by Occam's razor.

Significance. If correct, the paper provides a clean null result for this class of infrared-RG modified gravity: the CMB acoustic-peak structure together with joint distance and growth data confines the running parameter to |ν| ≲ 2×10^-4 at 2σ (taking Table 1 at face value), far too small to produce measurable changes in growth or expansion, and it overturns the earlier suggestion that this framework could alleviate the growth tension through RSD flexibility. The main qualitative conclusion is robust and sensible, and the paper is honest in reporting a negative result. The strengths are the fully action-based construction of the scale dependence, the closed-form analytic approximation of the ν-induced acoustic distortion (Eq. 52), the use of standard public data with a well-established pipeline and convergence checks, and the explicit acknowledgment that the claim is scoped to the adopted scale setting. The evident caveat is that the tight bound applies to the W = -2ψ scale identification inherited from Refs. [21, 27]; other scale-setting prescriptions are not constrained by this analysis, a point the paper itself concedes by saying 'as implemented here'.

major comments (3)
  1. [Section 6, Table 1] The central quantitative claim is not supported by the reported numbers. Section 6 states that the data 'allowed us to constrain the value of ν to |ν| ≲ 10−5', and the Introduction repeats that the value is 'of the order of 10−5'. However, Table 1 (joint CMB+BAO+SNIa+RSD) reports ν = 5.815×10^-5 with 2σ uncertainties +0.00017/−0.00017, i.e., ±1.7×10^-4, corresponding to a 95% interval of roughly [−1.1×10^-4, 2.3×10^-4]. The correct statement is |ν| ≲ 2.3×10^-4 at 2σ, or ν = (5.8 ± 8.5)×10^-5 at 1σ, which is within 0.7σ of zero; the bound in the text overstates the constraint by more than an order of magnitude and must be corrected. In addition, the caption of Table 1 says the limits are '2σ CL', yet the ΛCDM CMB-only value H0 = 67.26 ± 0.54 coincides numerically with the published Planck 2018 1σ value (67.36 ± 0.54 for TT,TE,EE+lowE), which suggests the quoted intervals may actually be 68% limits; the confidence-level convention must be verified and stated consistently, since the claim of compatibility 'within the 1σ confidence level' in Section 5.2 depends on it. The qualitative conclusion that the model does not resolve the tensions survives either reading, because even the 2σ upper bound produces sub-percent parameter shifts, but the headline constraint must be restated.
  2. [Appendix A, Eqs. (32), (68), (83)] The treatment of the logarithmic mode in the CDM initial conditions is delicate and under-specified. Equation (68) defines f(η) = 3ν ∫ ψ d ln η, and Eq. (83) gives δc containing the term 6νψ0 ln(η/η0); the text then states: 'In our numerical analysis we consider η ∼ η0, therefore, logarithmic term is canceled.' Because Eq. (32) contains the source term −6Hνψ, the Boltzmann integration regenerates a logarithmic contribution as the solution evolves away from the initial time, so the cancellation can only be a choice of the constant of integration, not an elimination of the mode. Since this contribution is first order in ν, which is exactly the order of the effect being constrained, the authors should specify how the CLASS implementation treats this term (whether the initial conditions contain the log and whether the source term is integrated consistently), and should demonstrate that the ν posterior is insensitive to the initial integration time and to η0. This is a corrigible but load-bearing point, and the paper's own text flags it as an assumption rather than a derivation.
  3. [Section 5.2] The modified CLASS code is not released. The paper states that the key equations are implemented in a 'modified version of the Boltzmann solver code CLASS' and cites line numbers (perturbations.c, lines 6500 and 9128), but no code or patch is provided. The central claim of the paper is a posterior distribution generated by this code, and the analytic equations alone do not pin down the delicate implementation choices discussed above, such as the handling of the logarithmic CDM mode and the choice of initial conditions. I request that the code or a patch be made publicly available as supplementary material at the revision stage so that the numerical result is reproducible.
minor comments (7)
  1. [Eq. (44)] Equation (44) contains a double minus sign and a missing prime: the friction term is written as 'Hδb' instead of 'Hδb′', and '− −4πG0' should be a single minus sign; the equation should read δb'' + Hδb′ − [4πG0/(1−ν)]a²(εcδc + εbδb) = 0.
  2. [Eq. (53)] The term ln(√3x/k) in the definition of K(x) is not dimensionally consistent as written, since √3x/k = √3rs has units of length; the argument of the logarithm should be a dimensionless ratio, and the reference scale should be specified.
  3. [Section 4 and near Eq. (16)] There are two typos: 'instroducing' should be 'introducing' in the text after Eq. (53), and 'strenght' should be 'strength' in the sentence defining ν below Eq. (16).
  4. [Section 5.2] The sentence 'we consider a total of seven cosmological parameters' is inconsistent with the footnote that adds the SN Ia absolute magnitude and high-ℓ CMB nuisance parameters; the sentence should distinguish the seven cosmological parameters from the additional nuisance parameters.
  5. [Section 3, text after Eq. (31)] The simplifying assumption that only CDM contributes to the background trace T (neglecting baryons) is asserted to be harmless; a one-line estimate of the fractional effect, which is of order νΩb/Ωm ≈ 0.2ν, would make this statement precise.
  6. [Figure 2] The red/orange distinction between the ΛCDM and RGGR contours is hard to discern in the printed rendering; distinct line styles or a zoomed inset would improve readability.
  7. [Section 6] Once the bound on ν is corrected, the conclusions should explicitly repeat that the bound applies to the W = −2ψ scale-setting (Eqs. 13–16); the text is already carefully scoped ('as implemented here'), but restating the conditionality next to the corrected number would prevent over-generalization in citations.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the constraint on nu is a fit to external CMB/BAO/SN/RSD data, and the self-cited framework Ref. [27] is model construction rather than a forced conclusion; separately, the text's '|nu| <~ 10^-5' bound is inconsistent with Table 1's 2-sigma errors, a numerical reporting issue.

full rationale

I walked the derivation chain from the action (Eq. 1) through the field equations (2)-(6), the scale-setting identification W = -2psi (Eqs. 13-16), the perturbed Einstein-Boltzmann system (25)-(33), the matter-dominated solutions (34)-(45), the acoustic-oscillator equation (47)-(53), the CLASS implementation, and the MCMC against Planck 2018, Pantheon, BAO, and RSD data. The reported constraint on the RGGR parameter nu is a fit to external datasets; nu never appears as a target inside the derivation. Every equation in the chain is obtained by varying the action or by linear perturbation theory, and none encodes the conclusion that nu is small: the smallness is an outcome of the data, as Fig. 1 shows that nu = +/-0.01 already displaces the CMB spectra substantially. The theoretical framework is inherited from the authors' prior work Ref. [27] (and Refs. [21, 29] for the scale setting), but this self-citation is not load-bearing for the central claim: the likelihoods used are independent of [27], and no result in [27] is invoked to forbid alternatives or to force the posterior. The paper is explicit that the scale setting is a choice ('There are different possible scale settings for cosmology', Sec. 2.3), so the constraint is properly conditional on that ansatz rather than circular. No fitted parameter is renamed as a prediction, and no equation reduces to its own input. One flagged inconsistency is numerical, not circular: Section 6 concludes '|nu| lesssim 10^-5', but Table 1 (CMB+BAO+SNIa+RSD) reports nu = 5.815e-5 with 2-sigma errors of +/-1.7e-4, i.e., a 95% interval of roughly [-1.1e-4, 2.3e-4]; the 1-sigma error is 8.5e-5, so nu = 0 is within 1 sigma as the paper claims, but nu can be as large as 2.3e-4, over an order of magnitude above the stated bound. This overstatement should be corrected, but it does not change the qualitative conclusion that the model cannot resolve the H0 and sigma8 tensions. Overall circularity is minimal: score 1.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The central claim rests on the RGGR action and scale-setting assumptions from prior work by the same group, plus a linear expansion in the single free parameter nu. No new physical entities beyond the auxiliary field gamma and the RG-scale bookkeeping are introduced. The observational constraints themselves come from public external data, so the ledger is dominated by theoretical priors rather than hidden fitted parameters.

free parameters (4)
  • nu (RGGR running parameter) = (5.815e-05)^{+0.00017}_{-0.00017} (2 sigma, joint); 8.372e-05 +/- 1.8e-04 (CMB only)
    The dimensionless parameter controlling the running of G and Lambda with the perturbation W; constrained by MCMC to be consistent with zero.
  • Standard LambdaCDM parameters {omega_b, omega_cdm, n_s, H_0, sigma_8, tau_reio} = Values in Table 1 (e.g., H0=67.86, sigma8=0.8089 joint)
    The six baseline cosmological parameters fitted simultaneously; not ad hoc, but their errors affect the nu constraint via degeneracies.
  • SN Ia absolute magnitude M = Not quoted
    Nuisance parameter in the Pantheon likelihood (Sec. 5.1).
  • Integration constant eta0 in CDM initial condition = Set so eta is approximately eta_i, cancels log term
    Appears in delta_c initial condition (eq. 83); the choice removes the logarithmic growing term. It is not a cosmological parameter but a boundary-condition choice.
assumptions (5)
  • domain assumption The effective action (1), with Lagrange multipliers enforcing scale settings mu_p = f_p, correctly describes infrared RG corrections to gravity.
    The entire theoretical framework is assumed from Refs [27,29,21]; no derivation from a fundamental quantum-gravity theory is given.
  • ad hoc to paper The first RG scale is set by W = U^alpha U^beta (g_alpha_beta - gamma_alpha_beta) = -2 psi, i.e., tied to the Newtonian potential perturbation.
    Eqs. (13)-(14) in Sec. 2.3. This scale-setting choice determines the nu dependence of all perturbation equations.
  • ad hoc to paper G and Lambda run linearly in W: G0 G^{-1}(W) = 1 + nu W + O(W^2), and Lambda approx Lambda0 + (Lambda0 - 4 pi G0 T0) delta G.
    Eqs. (16)-(17). The truncation at first order defines the single parameter nu and the Lambda-G relation; higher-order terms are neglected.
  • domain assumption Only cold dark matter contributes to the anomalous energy-momentum transfer Q_beta; baryon contributions are negligible.
    Stated in Sec. 3: 'here we assume that only dark matter contributes to the background value of T. This is a simplifying assumption.'
  • domain assumption Adiabatic initial conditions with phi = constant (phi0) at early times are the physically relevant branch of the perturbation solutions.
    Appendix A, eqs. (75)-(81). The other solutions are decaying or singular as nu approaches 0, so only the constant mode is retained.
invented entities (2)
  • Auxiliary tensor field gamma_alpha_beta
    purpose: Sets the RG-effects background metric: gamma_alpha_beta = a^2(eta) eta_alpha_beta (eq. 12), used to define W and to keep the background field equations identical to GR.
    Non-dynamical auxiliary field appearing in the action (1) and in the definition of W (eq. 14). It has no independent observable signature and is eliminated in the background limit.
  • RG scales mu_p and Lagrange multipliers lambda_p
    purpose: Bookkeeping for scale-dependence in the action; mu_p can be eliminated in favor of f_p, so they are not independent physical fields.
    Introduced in action (1); eq. (3) identifies mu_p with f_p. The paper notes mu_p is not physically essential.

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Cite this review

Pith. "Pith review of Scale-dependent and background-preserving gravity from an action: cosmological tests." pith.science (2026). https://pith.science/paper/YYPIJ6T2

@misc{pith2026241112097,
  author       = {Pith},
  title        = {Pith review of: Scale-dependent and background-preserving gravity from an action: cosmological tests},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YYPIJ6T2}},
  note         = {Machine review of arXiv:2411.12097}
}
abstract

We investigate the observational implications of a gravitational model wherein the gravitational constant $G$ and the cosmological constant $\Lambda$ exhibit scale-dependent behavior at the perturbative level, while preserving the General Relativity (GR) field equations at the background. This model is motivated by the potential influence of large-scale (infrared) Renormalization Group (RG) corrections to gravity and is constructed upon an effective action incorporating a scale definition via Lagrange multipliers. We explore the effects of these modifications during the recombination epoch with particular focus on their impact on the structure of acoustic oscillations. Additionally, we perform a comprehensive parameter fitting analysis using data from the Cosmic Microwave background (CMB), type Ia Supernovae (SN Ia), Baryon Acoustic Oscilations (BAO) and Redshift Space Distortions (RSD). Our results indicate that the RG corrections here considered are consistent with the main predictions of the $\Lambda$CDM model, and they slightly increase the uncertainties in the parameter estimations. Such small differences cannot be used to dismiss the current cosmological tensions. Although previous results indicated that this model is more flexible than $\Lambda$CDM regarding RSD data, potentially alleviating tensions, this advantage becomes negligible with the current extended data set. The framework maintains its theoretical consistency and foundation; however, unless further generalized, it cannot effectively address current cosmological issues.

Figures

Figures reproduced from arXiv: 2411.12097 by the authors.

Figure 1
Figure 1. CMB angular power spectrum for GR (ν = 0), and RGGR case with ν = 0.01, and ν = −0.01. where a scale-invariant primordial spectrum was used, jl represents spherical Bessel functions, k denotes the Fourier modes and η0 indicates the current conformal time. Inte￾grating eq. (54) in the GR case yields an almost constant Dℓ (a low-scales plateau). Clearly, from eq. (37), we ob￾serve that RGGR causes an upward or downwar… view at source ↗
Figure 2
Figure 2. Analysis with data from RSD. In red we have the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Statistical analysis of the joint analysis with data from CMB, BAO, SNIa and RSD. In red we have the results for [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

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