Pith. sign in

REVIEW 4 major objections 4 minor 31 references

Scale dependence of the effective gravitational constant from functional renormalization group

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that the scale dependence of the effective gravitational constant is governed by one universal functional-RG equation, with two regulator-dependent parameters that existing observations pin into a tiny window—effectively…

desk verdict A clean ODE analysis of a two-parameter running-G model, but the paper's claim that the same equation holds with matter or a cosmological constant is asserted rather than derived, and the observational constraints are oversold. read the letter →

arxiv 2608.07875 v1 pith:A2KLBCWL submitted 2026-08-08 gr-qc hep-th

Ruiqi Liang , Zhoujian Cao , Bing Sun This is my paper · ORCID
classification gr-qchep-th
keywords scale-dependentgravitationalconstantfunctionalrenormalizationgroupanomalousdimensionregulatordependencevaryingGwaveconstraintscosmologicaleffectiveNewton
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 tries to establish that the scale dependence of the effective gravitational constant is governed by one master renormalization-group equation, no matter whether the theory contains only gravity, also a cosmological constant, also electromagnetism, or all three. The equation is $\partial_t g_k = (2+\eta_N) g_k$ with $\eta_N = g_k B_1(0)/(1-g_k B_2(0))$, where $t=\ln(k/k_*)$ and $B_1(0), B_2(0)$ are two regulator-shape constants. The authors treat those constants as free and show that demanding every scale be physically reachable restricts them to certain 'advisable' regions. They then map the RG scale to distance, time, curvature, and temperature and compare with existing bounds, concluding that the allowed window is so narrow that the effective gravitational constant is observationally constant. A sympathetic reader would care because this sharpens what quantum-gravity running of $G$ would have to look like for any future detection of a varying $G$.

What carries the argument

The load-bearing object is the graviton anomalous dimension $\eta_N$ inside the functional RG flow, expressed through two regulator-shape constants $B_1(0)$ and $B_2(0)$. The master equation $\partial_t g_k=(2+\eta_N)g_k$ does all the work: the paper asserts its form is unchanged when photon or cosmological-constant sectors are added, so the closed-form solution for $g_k(t)$—and hence all later statements about limit scales, admissible parameter regions, and observational constraints—follows from this single equation plus an identification of $k$ with a physical scale (distance, time, curvature, or temperature).

What would settle it

A first-principles FRG calculation with dynamical photons and a cosmological constant that shows $\eta_N$ picks up terms proportional to the fine-structure constant or to $\lambda_k$ would falsify the claimed universality of Eq. (4). Observationally, any detected time drift $|\dot G/G_N| \gtrsim 10^{-14}\,\mathrm{yr}^{-1}$, or a change of effective $G$ larger than about $20\%$ between a gravitational-wave source and observer separated by 40 Mpc, would violate the paper's central bound.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the running of the effective Newton constant is always described by the same equation, $\partial_t g_k = (2+\eta_N)g_k$ with $\eta_N = g_k B_1(0)/\bigl(1-g_k B_2(0)\bigr)$, irrespective of whether electromagnetic interactions or a cosmological constant are included. Since the regulator is not fixed by first principles, $B_1(0)$ and $B_2(0)$ are treated as free parameters, and the paper analyzes the full solution in the $B_1$–$B_2$ plane. Requiring that no intermediate scale be singular (so $g_k$ never hits $1/B_2(0)$) selects three admissible regions. Imposing existing observational limits—GW170817's $\lesssim 20\%$ bound on $G$ changes over 40 Mpc, the recombination-era $\lesssim 5\%$ bound, the torsion-balance bound $|\dot G/G_N|<10^{-14}\,\mathrm{yr}^{-1}$, and lunar-laser-ranging $|\ddot G/G_N|<10^{-15}\,\mathrm{yr}^{-2}$—the paper concludes that only $B_2(0)<0$ or $|B_1(0)|\lesssim 10^{-42}$ survives. Thus the effective gravitational constant is, for all practical observations, constant.

Load-bearing premise

The whole analysis rests on the claim that the anomalous dimension $\eta_N$ keeps exactly the same functional form when electromagnetic fields or a cosmological constant are present; if those extra sectors contribute to $\eta_N$, then the master equation (4) is not universal and the derived parameter constraints do not follow.

Editorial extensions

If this is right

  • Within this framework, the form of the running of $G$ cannot be changed by adding standard-model matter: electromagnetism and cosmological-constant-type interactions leave the master equation untouched.
  • The observables the paper uses—GW170817 propagation, recombination-era cosmology, torsion balances, and lunar laser ranging—force any viable FRG model of varying $G$ into either $B_2(0)<0$ or $|B_1(0)|\lesssim 10^{-42}$, so the effective gravitational constant is indistinguishable from a constant over all probed scales.
  • All explicit parameter choices plotted in the paper, including the regulator $R^{(0)}(z)=z/(e^z-1)$, fall outside the allowed region, so those specific RG-improved models are observationally excluded.
  • If $B_2(0)>0$ and $B_1(0)$ lies in the wrong range, the solution hits a finite limit scale where $g_k\to 1/B_2(0)$; the 'all scales accessible' assumption converts this into a physical cutoff and rules those parameters out.
  • The GW170817-based bound $|B_1(0)|\lesssim 10^{-42}$ is the strictest of the scale identifications considered; distance-based identifications constrain the model far more tightly than temperature- or curvature-based ones.

Reading between the lines

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

  • The claimed universality of $\eta_N$ is fragile: in many FRG treatments matter loops do feed into the graviton anomalous dimension, so a dedicated matter-including calculation could readily overturn Eq. (4) even if the paper's internal algebra is correct.
  • The spread between the bounds obtained from different scale identifications ($10^{-42}$ for distance versus $10^{-4}$ for curvature) suggests the physical meaning of the RG scale is underdetermined; comparing several observables at once could discriminate among identifications.
  • Reading the constraints as a statement about regulators, the allowed window can be interpreted as selecting regulators with $B_2(0)<0$; if one trusts the framework, the regulator is the only free input, and the observational data effectively chooses it.
  • If the bound $|B_1(0)|\lesssim 10^{-42}$ is taken seriously, RG-improved black-hole models that invoke a strongly running $G$ to resolve singularities would need to confine all running to scales far above astrophysical ones, weakening the usual motivation for such constructions.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the functional renormalization group flow of the dimensionless Newton constant g_k = k^2 G_k and claims that its scale dependence is always governed by the single equation ∂_t g_k = (2 + η_N)g_k with η_N = g_k B_1(0)/(1 - g_k B_2(0)), regardless of whether a cosmological constant or electromagnetic interactions are included. It solves this ODE, derives 'advisable' regions in the (B_1(0), B_2(0)) plane under the assumption that all scales are physically accessible, and then uses multiple scale identifications (distance, time, curvature, temperature) to turn observations such as GW170817 and cosmological bounds into strict constraints on B_1(0) and B_2(0), concluding that effectively B_2(0) < 0 or |B_1(0)| < 10^-42.

Significance. If the claimed universality of Eq. (4) were established, the paper would provide a useful closed-form characterization of the possible scale dependence of G and a transparent way to translate observations into constraints on the regulator-dependent parameters. The algebraic solution of the ODE and the parameter-region analysis are explicit and internally consistent, and the attempt to confront the running with independent observations is a genuine strength. However, the central premise -- that η_N is independent of λ_k and of matter fields -- is asserted rather than derived, and it conflicts with standard gravity-matter FRG results. As a result, the significance of the observational constraints is currently not established.

major comments (4)
  1. [II, Eqs. (12)-(13)] The central reduction of the coupled system to the single equation (4) is not derived. In the Bonanno-Reuter system that the paper adopts, the anomalous dimension is η_N = g_k B_1(-2λ_k)/(1 - g_k B_2(-2λ_k)) with B_1 and B_2 defined in Eqs. (6)-(7); Eq. (13) itself contains λ_k-dependent threshold functions. Eq. (5) instead uses B_1(0) and B_2(0), and the paper never shows that λ_k decouples from η_N. Consequently the solution (29) and every constraint derived from it in Section IV are not consequences of the actual coupled flow unless an additional, unstated approximation is imposed.
  2. [II, Eqs. (14)-(20)] The claim that electromagnetic or Yang-Mills interactions do not affect the running of the gravitational constant is asserted rather than derived. In a standard single-metric gravity-matter FRG, matter loops enter the graviton two-point function and hence η_N; the displayed system leaves η_N unchanged while giving matter its own flow (Eqs. (15)-(16) and (18)-(20)). The paragraph after Eq. (20), stating that extra interactions do not affect the more fundamental running couplings, therefore holds only because η_N was written without matter contributions. This is load-bearing because the abstract and conclusion present the universality of Eq. (4) as the main result.
  3. [IV.A-IV.D] The observational constraints are not robust because the identification of the RG scale k is an additional free input. The same observed limits are mapped to |B_1(0)| ≲ 10^-42 (distance), ≲ 10^-36 (time), ≲ 10^-4 (Kretschmann curvature), with different t values for each convention, and no physical principle is given for preferring one identification. The headline conclusion in Section V that either B_2(0) < 0 or |B_1(0)| < 10^-42 is therefore an artifact of one particular scale-setting convention.
  4. [V] The Conclusion states that the regulator function, and hence B_1 and B_2, 'may be affected by the interactions being considered.' This admission conflicts with the earlier claim that the scale dependence of G is independent of other interactions, because a change in B_1 or B_2 changes the quantitative running even if the form of Eq. (4) is kept. The internal tension should be resolved before the universality claim can be accepted.
minor comments (4)
  1. [II, Eq. (3)] The definition α_k ≡ α_k appears to be a typo; the scale-dependent fine-structure constant should be defined nontrivially or the identity is content-free.
  2. [II, Eq. (11)] The displayed expression for η_N with regulator (10) should be checked against Eqs. (5)-(7) and (42)-(43); substituting (42)-(43) into (5) gives η_N = (π/18 - 8/π)g_k/(1 - 2g_k/(3π)), which does not match the expression as printed.
  3. [III, Fig. 6 caption] The caption quotes B_1(0) ≈ -0.78, while Eq. (42) gives B_1(0) = π/18 - 8/π ≈ -2.37; this discrepancy should be resolved.
  4. [IV.A] The GW170817 bound on ΔG/G is attributed to Refs. [14-16] rather than the original LIGO/Virgo data release; a direct reference would aid verification.

Circularity Check

2 steps flagged · score 6.0 of 10

Universality claim is an ansatz: λ-dependent threshold functions are frozen at y=0, and the EM running uses the gravity-only η_N by definition; the parameter-solution and data constraints are independent once Eq. (4) is granted.

  1. ansatz smuggled in via citation [Sec. II, after Eqs. (12)-(13)]
    "When gk and λk are considered which takes both the gravitational interaction and the cosmological constant involved interactions into consideration, we have the renormalization group equation [8] ∂tgk = (2 + ηN ) gk ... We can find out that Eq. (4) is exactly the same to Eq. (12). That is to say no matter we consider the cosmological constant involved interactions or not, the resulting scale dependence of G is the same."

    The functions B1(y),B2(y) in (6)-(7) depend on y, and the coupled FRG contains thresholds at -2λ. In the cited Bonanno-Reuter system ηN = gB1(-2λ)/(1-gB2(-2λ)), so Eq. (12) equals Eq. (4) only if the thresholds are frozen at λ=0. The paper supplies no derivation of this decoupling; it writes ηN with B1(0),B2(0) in (5) and then declares the equations the same. Section III solves Eq. (4) with constant B1(0),B2(0) for all cases, so the conclusion 'scale dependence of G is the same' is equivalent to the input assumption that threshold functions are evaluated at y=0.

  2. ansatz smuggled in via citation [Sec. II, after Eqs. (14)-(16)]
    "If both the gravitational interaction and the electromagnetic interaction are considered, both gk and αk should be involved, and the resulting renormalization group equation reads as ∂tgk = (2 + ηN )gk ... Again we find out that Eq. (4) is exactly the same to Eq. (14). That is to say the electromagnetic interaction does not affect the resulting scale dependence of G."

    No graviton two-point function with photon loops is evaluated; the same symbol ηN is imported from the gravity-only equation. Eq. (14) is formally identical to Eq. (4) by construction because ηN is reused unchanged, so the statement that electromagnetism does not affect G is not a prediction but a restatement of the assumption. The later observational constraints in Sec. IV inherit this assumption when Eq. (29), derived from constant B1(0),B2(0), is applied to a universe with electromagnetic interactions.

full rationale

The algebraic solution of Eq. (4), the advisable parameter regions of Sec. III, and the observational constraints of Sec. IV are internally self-contained once Eq. (4) is granted; comparisons with GW170817, cosmological bounds, and lunar laser ranging are independent external data, and the authors' self-citations [14-16] rest on public LIGO observations. Those parts are not circular. However, the paper's central new claim—that adding a cosmological constant or an electromagnetic field leaves the running of G governed by exactly the same equation with the same two constants B1(0),B2(0)—is not derived. The threshold functions in (6)-(7) carry arguments, and in the cited Bonanno-Reuter and Reuter systems the graviton anomalous dimension is evaluated at -2λ (and, in standard single-metric FRG with matter, receives matter contributions). The paper writes Eq. (5) with B1(0),B2(0) and then asserts that Eq. (4) is 'exactly the same' as Eq. (12) and Eq. (14); that equality is by construction, not by calculation. Since Sec. III solves Eq. (4) with constant B1(0),B2(0) for all interaction contents and Sec. IV uses those solutions to constrain parameters, the universality result reduces to the unstated ansatz that ηN is independent of λ and matter. This warrants a partial circularity score of 6: the predicted universality is equivalent to its input, while the parameter-region and observational analysis retain independent content.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The derivation leans on the standard FRG framework and the Einstein-Hilbert truncation, neither of which is derived here. The decisive extra input is the unproven decoupling of G's running from matter and λ, plus the ad hoc scale identifications. These assumptions, rather than the algebraic manipulation, bear the weight of the conclusions.

free parameters (3)
  • B1(0) = |B1(0)| < 1e-42 (distance identification) or < 1e-8 (temperature identification)
    Regulator-dependent threshold constant in the gravitational anomalous dimension; cannot be determined from first principles in this truncation.
  • B2(0) = not directly constrained; 'all scales accessible' excludes B2(0)=1; e.g., regulator (10) gives B2(0) ≈ 0.21
    Regulator-dependent threshold constant controlling the divergence denominator; the requirement that g_k never reaches 1/B2(0) defines the advisable regions.
  • scale identification choice = one of inverse distance, inverse time, K^1/4, temperature
    The mapping from RG scale k to physical scale is chosen ad hoc in Section IV; each choice gives different constraints on B1 and B2.
assumptions (6)
  • standard math Functional renormalization group and Wetterich equation provide the exact flow of the effective average action.
    Used implicitly in Section II to derive the beta functions.
  • domain assumption The effective action can be truncated to the Einstein-Hilbert term plus cosmological constant and abelian gauge term.
    The entire analysis is confined to this truncation; higher-order curvature operators are neglected.
  • ad hoc to paper The anomalous dimension of the gravitational coupling is unaffected by matter fields and by the cosmological constant.
    Stated in Eqs. (12), (14), (17) without derivation; central to the claim that Eq. (4) governs all cases. Contradicts standard gravity-matter FRG results.
  • domain assumption All energy scales are physically accessible, so g_k must never reach the divergence 1/B2(0).
    Used in Section III to define 'advisable' parameter regions; if a limit scale exists, the regions would change.
  • ad hoc to paper The RG scale k can be identified with inverse distance, inverse time, K^1/4, or temperature.
    Each identification in Section IV is a conjecture; different choices give incompatible bounds on B1.
  • domain assumption Observational constraints (GW170817, CMB recombination, lunar laser ranging, torsion balance) apply to G at the scales associated with those experiments.
    The paper cites these bounds and interprets them as constraints on G_k at specific t values.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Scale dependence of the effective gravitational constant from functional renormalization group." pith.science (2026). https://pith.science/paper/A2KLBCWL

@misc{pith2026260807875,
  author       = {Pith},
  title        = {Pith review of: Scale dependence of the effective gravitational constant from functional renormalization group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A2KLBCWL}},
  note         = {Machine review of arXiv:2608.07875}
}
read the original abstract

The general relativity may not be the final theory of gravity. One possible way to look for the gravity theory beyond general relativity is considering variable gravitational constant. Based on the quantum field theory, the gravitational constant as the coupling constant of gravity interaction may change along with the energy scale. Such scale dependent behavior of the gravitational constant can be well described by the functional renormalization group. In the current work, we investigate such behavior systematically. Firstly we find that such behavior is qualitatively independent of interactions such as the electromagnetic interaction included or not. But quantitatively the behavior is governed by two to-be-determined parameters. In general a limit scale may be introduced by the scale dependence of the gravitational constant. If we assume all physical scales are feasible, the two parameters are limited in some special regions. And more we compare the scale dependence behavior to the existing observations. We find the observation results constraint the two parameters strictly.

Figures

Figures reproduced from arXiv: 2608.07875 by the authors.

Figure 1
Figure 1. FIG. 1: The advisable regions (marked with ‘advisable’) for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Scale dependence of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 3
Figure 3. The IR limit of G is very large. The behavior of G and gk is similar to [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 6
Figure 6. Figure 6: FIG. 6: Scale dependence of [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Scale dependence of [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Observation constraint of parameters [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Observation constraint of parameters [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 28 canonical work pages

  1. [1]

    Regular Black Holes: A Short Topic Review

    Chen Lan, Hao Yang, Yang Guo, and Yan-Gang Miao. Regular Black Holes: A Short Topic Review. Interna- tional Journal of Theoretical Physics , 62(9):202, Septem- ber 2023

    work page 2023

  2. [2]

    Transfer time of black hole-white holes

    Shiyue Ren and Zhoujian Cao. Transfer time of black hole-white holes. Communications in Theoretical Physics, 2026

    work page 2026

  3. [3]

    Covariant dynamics from static spherically symmetric geometries

    Cong Zhang and Zhoujian Cao. Covariant dynamics from static spherically symmetric geometries. Phys. Rev. Lett., 135:261401, Dec 2025

    work page 2025

  4. [4]

    Donoghue

    John F. Donoghue. Introduction to the Effective Field Theory Description of Gravity. arXiv e-prints , pages gr– qc/9512024, December 1995

    arXiv 1995

  5. [5]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J.M. Pawlowski, M. Tissier, and N. Wschebor. The nonper- turbative functional renormalization group and its appli- cations. Physics Reports, 910:1–114, 2021

    work page 2021

  6. [6]

    M. Reuter. Nonperturbative evolution equation for quan - tum gravity. Phys. Rev. D , 57:971–985, Jan 1998

    work page 1998

  7. [7]

    Bonanno and M

    A. Bonanno and M. Reuter. Quantum gravity effects near the null black hole singularity. Phys. Rev. D , 60:084011, Sep 1999

    work page 1999

  8. [8]

    Renormalization group improved black hole spacetimes

    Alfio Bonanno and Martin Reuter. Renormalization group improved black hole spacetimes. Phys. Rev. D , 62:043008, Jul 2000

    work page 2000

Show all 31 references
  1. [9]

    J.-E. Daum, U. Harst, and M. Reuter. Running gauge coupling in asymptotically safe quantum gravity. Journal of High Energy Physics , 2010:84, January 2010

    2010

  2. [10]

    Structural aspect s of asymptotically safe black holes

    Benjamin Koch and Frank Saueressig. Structural aspect s of asymptotically safe black holes. Classical and Quan- tum Gravity, 31(1):015006, nov 2013

    2013

  3. [11]

    Alfio Bonanno, Mariano Cadoni, Mirko Pitzalis, and An- drea P. Sanna. Effective quantum spacetimes from func- tional renormalization group. Phys. Rev. D , 111:064031, Mar 2025

    2025

  4. [12]

    The fundamental constants and their variation: observational and theoretical status

    Jean-Philippe Uzan. The fundamental constants and their variation: observational and theoretical status. Rev. Mod. Phys. , 75:403–455, Apr 2003

    2003

  5. [13]

    Varying constants, gravitation a nd cosmology

    Jean-Philippe Uzan. Varying constants, gravitation a nd cosmology. Living Reviews in Relativity , 14(1):2, March 2011

    2011

  6. [14]

    The effect of the gravitational constant varia- tion on the propagation of gravitational waves

    Jiachen An, Yadong Xue, Zhoujian Cao, Xiaokai He, and Bing Sun. The effect of the gravitational constant varia- tion on the propagation of gravitational waves. Physics Letters B , 844:138108, September 2023

    2023

  7. [15]

    Constrain the time variation of the gravitational constant via the propagation of gravitational waves

    Bing Sun, Jiachen An, and Zhoujian Cao. Constrain the time variation of the gravitational constant via the propagation of gravitational waves. Physics Letters B , 848:138350, 2024

    2024

  8. [16]

    The effect of the gravitational constant variation on the phase of grav- itational waves

    Jiachen An, Bing Sun, and Zhoujian Cao. The effect of the gravitational constant variation on the phase of grav- itational waves. Physics of the Dark Universe , 50:102062, 2025

    2025

  9. [17]

    Loop quantum gravity and cosmological constant

    Xiangdong Zhang, Gaoping Long, and Yongge Ma. Loop quantum gravity and cosmological constant. Physics Let- ters B , 823:136770, December 2021

    2021

  10. [18]

    Exact evolution equation for the e f- fective potential

    Christof Wetterich. Exact evolution equation for the e f- fective potential. Physics Letters B , 301(1):90–94, Febru- ary 1993

    1993

  11. [19]

    The Wetterich Equation , pages 17–

    Stefan Fl¨ orchinger. The Wetterich Equation , pages 17–

  12. [20]

    Black Holes in Asymptotically Safe Gravity

    Alessia Platania. Black Holes in Asymptotically Safe Gravity. arXiv e-prints, page arXiv:2302.04272, February 2023

    2023 arXiv

  13. [21]

    Springer Berlin Heidelberg, Berlin, Heidelberg, 2010

    2010

  14. [22]

    Black Holes in Asymptotic Safety: A Re- view of Solutions and Phenomenology

    Andrea Spina. Black Holes in Asymptotic Safety: A Re- view of Solutions and Phenomenology. arXiv e-prints , page arXiv:2510.14552, October 2025

    2025

  15. [23]

    On the cutoff scale identification of flrw cosmology in asymptotically safe gravity

    Chiang-Mei Chen, Rituparna Mandal, and Nobuyoshi Ohta. On the cutoff scale identification of flrw cosmology in asymptotically safe gravity. Classical and Quantum Gravity, 42(9):095011, apr 2025

    2025

  16. [24]

    E. G. Adelberger, J. H. Gundlach, B. R. Heckel, S. Hoedl, and S. Schlamminger. Torsion balance experiments: A low-energy frontier of particle physics. Progress in Par- ticle and Nuclear Physics , 62(1):102–134, January 2009

    2009

  17. [25]

    Bonanno and M

    A. Bonanno and M. Reuter. Cosmology of the planck era 9 from a renormalization group for quantum gravity. Phys. Rev. D , 65:043508, Jan 2002

    2002

  18. [26]

    Wmap constraints on scalar-tensor cosmology and the variation of the gravitational constant

    Ryo Nagata, Takeshi Chiba, and Naoshi Sugiyama. Wmap constraints on scalar-tensor cosmology and the variation of the gravitational constant. Phys. Rev. D , 69:083512, Apr 2004

    2004

  19. [27]

    Variations of the gravitational constant from lunar laser ranging data

    Jurgen Muller and Liliane Biskupek. Variations of the gravitational constant from lunar laser ranging data. Classical and Quantum Gravity , 24(17):4533, aug 2007

    2007

  20. [28]

    Merkowitz

    Stephen M. Merkowitz. Tests of gravity using lunar lase r ranging. Living Reviews in Relativity , 13(1):7, Nov 2010

    2010

  21. [29]

    Asymp- totic safety casts its shadow

    Aaron Held, Roman Gold, and Astrid Eichhorn. Asymp- totic safety casts its shadow. Journal of Cosmology and Astroparticle Physics, 2019(6):029, June 2019

    2019

  22. [30]

    Ostriker

    Joachim Wambsganss, Renyue Cen, Guohong Xu, and Jeremiah P. Ostriker. Effects of weak gravitational lens- ing from large-scale structure on the determination of q0. The Astrophysical Journal , 475(2):L81, feb 1997

    1997

  23. [31]

    Planck 2018 results

    N Aghanim et al. Planck 2018 results. vi. cosmological parameters. Astron. Astrophys, 641:A6, 2020

    2018

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.