Scale-Dependent Gravitational Constant

• Scale-Dependent Effective Gravitational Constant is a theoretical framework where Newton’s constant varies with length, momentum, or spacetime scales due to renormalization group and quantum effects.
• The approach modifies the Einstein-Hilbert action by allowing both G and Λ to run with a covariantly set RG scale, influencing cosmological evolution and local gravitational phenomena.
• Observational constraints from CMB, BAO, SN Ia, RSD, and strong lensing tightly limit deviations from GR, while future surveys aim to further probe these scale-dependent effects.

A scale-dependent effective gravitational constant is a theoretical construct in which Newton’s "constant" $G$ is promoted to a function of length, momentum, or spacetime scales, motivated by renormalization group (RG) effects, quantum corrections, or new fields in extended gravity theories. Rather than being strictly constant, $G$ acquires non-trivial running with respect to a physical scale, leading to observable modifications in the behavior of gravity across cosmic and astrophysical environments. This scale-dependence can be parametrized in various frameworks (RG-improved actions, scalar-tensor theories, effective field theory), and is increasingly constrained by multi-probe cosmological datasets and gravitational experiments.

1. Theoretical Formulation and Covariant Scale-Setting

The action principle underlying scale-dependent gravity generalizes the Einstein-Hilbert action by allowing both $G$ and the cosmological constant $\Lambda$ to depend on an RG scale, typically denoted $\mu$, which itself is tied to physical quantities such as the Hubble parameter $H$, local curvature invariants, or combinations of geometric and matter fields. A generic action takes the form

$$S[g,\mu,\lambda,\Psi,\gamma] = S_m[g,\Psi] + \frac{1}{16\pi} \int d^4x\,\sqrt{-g}\left\{ \frac{R-2\Lambda(\mu)}{G(\mu)} + \sum_p \lambda_p [\mu_p - f_p(g,\Psi,\gamma)] \right\}$$

where $\lambda_p$ are Lagrange multipliers enforcing scale-setting constraints, and $\gamma_{\alpha\beta}$ is an auxiliary reference metric (often the FLRW background). The scale-setting is fixed variationally: varying with respect to $\lambda_p$ enforces $G$0 and varying with respect to $G$1 provides consistency conditions relating the running of $G$2 and $G$3 to the underlying field content and spacetime geometry (Hipólito-Ricaldi et al., 2024, Bertini, 2024).

For cosmological and post-Newtonian backgrounds, the RG scale is typically covariantly related to $G$4, local energy density, or curvature, e.g., $G$5 in FLRW or $G$6 in general irrotational fluids (Bertini, 2024, Bertini et al., 6 Aug 2025). The requirement of energy-momentum conservation further constrains the permissible forms of running and relates the beta-functions of $G$7 and $G$8.

2. Phenomenology in Cosmology: Background and Perturbative Effects

Background Cosmology

On cosmological backgrounds, the scale dependence of $G$9 manifests in the modified Friedmann equations. Retaining $G$0-dependence at the perturbative level while ensuring standard $G$1CDM expansion at the background is central to the RGGR model (Hipólito-Ricaldi et al., 2024): $G$2 In many viable scenarios, the running is taken as logarithmic or affected by loop-induced corrections: $G$3 with $G$4 constrained to $G$5–$G$6 by data (Hernández-Arboleda et al., 2018, Alvarez et al., 2022, Bertini, 2024). One-loop quantum gravity computations fix the unique relation $G$7, but consistent covariant cosmologies often require an additional running scale associated with higher-derivative curvature terms to maintain energy-momentum conservation (Bertini et al., 2024).

Linear and Nonlinear Perturbations

At the linear cosmological perturbation level, $G$8 and $G$9 may acquire explicit dependence on perturbed quantities, e.g., Newtonian-gauge potential $\Lambda$0. For subhorizon modes $\Lambda$1, the effective gravitational constant becomes

$\Lambda$2

and the modified growth equation for the matter contrast reads

$\Lambda$3

(Hipólito-Ricaldi et al., 2024). On horizon and superhorizon scales, scale-dependent corrections manifest through modifications to the temporal and spatial parts of the perturbation equations, leading to distinct features in the anisotropic stress, gravitational slip ($\Lambda$4), and the integrated Sachs-Wolfe effect.

In general EFT approaches, scale-dependent corrections to $\Lambda$5 appear as series in $\Lambda$6, where $\Lambda$7 is a new gravitational mass scale. For $\Lambda$8, corrections are suppressed as $\Lambda$9; departures become significant only if $\mu$0 is well below the Hubble scale, i.e., $\mu$1 Mpc (Baker et al., 2014).

3. Astrophysical and Local Regimes

In non-cosmological, spherically symmetric, or weak-field settings, the RG scale is typically identified with local geometric or matter variables, such as $\mu$2. The effective action then yields field equations in which derivatives of $\mu$3 act as effective source terms: $\mu$4 with

$\mu$5

(Contreras et al., 2018). Such running can source self-sustained wormhole geometries in vacuum—i.e., without exotic matter content—with the throat size tied directly to the behavior of $\mu$6. The running $\mu$7 can in principle grow unbounded near the Planck scale, providing a concrete mechanism for UV modifications of spacetime geometry.

In Solar System and post-Newtonian contexts, scale dependence induces corrections at the first post-Newtonian (1PN) level via new source potentials, e.g., $\mu$8 satisfying $\mu$9. However, these corrections renormalize only internal structure and do not affect center-of-mass orbital motion; thus, standard PPN parameters (e.g., $H$0, $H$1) remain as in GR and Solar System tests do not constrain the running (Bertini et al., 6 Aug 2025).

4. Observational Constraints and Cosmological Signatures

Cosmological Probes

Constraints are primarily set by high-precision measurements of:

• CMB temperature, polarization, and lensing spectra (e.g., Planck 2018)
• Baryon Acoustic Oscillations (BAO)
• Type Ia Supernovae (SN Ia)
• Redshift Space Distortions (RSD)
• Weak lensing surveys

In action-based RGGR with perturbative running, joint analyses of CMB+BAO+SN+RSD data yield $H$2 and bound $H$3 (Hipólito-Ricaldi et al., 2024). Even in frameworks allowing richer $H$4-dependence, current Stage-4 survey forecasts suggest that transition scales below a few hundred Mpc (i.e., $H$5 Mpc) are required for detection (Baker et al., 2014).

For running vacuum models,

$H$6

is enforced by the combination of datasets spanning $H$7, supernovae, and BAO (Hernández-Arboleda et al., 2018, Alvarez et al., 2022). Rates of change $H$8–$H$9 are consistent with current bounds.

Astrophysical and Small-Scale Probes

Strong lensing analyses allow direct constraints on the scale-dependent gravitational slip parameter $$S[g,\mu,\lambda,\Psi,\gamma] = S_m[g,\Psi] + \frac{1}{16\pi} \int d^4x\,\sqrt{-g}\left\{ \frac{R-2\Lambda(\mu)}{G(\mu)} + \sum_p \lambda_p [\mu_p - f_p(g,\Psi,\gamma)] \right\}$$0, which is related to $$S[g,\mu,\lambda,\Psi,\gamma] = S_m[g,\Psi] + \frac{1}{16\pi} \int d^4x\,\sqrt{-g}\left\{ \frac{R-2\Lambda(\mu)}{G(\mu)} + \sum_p \lambda_p [\mu_p - f_p(g,\Psi,\gamma)] \right\}$$1 via the ratio of two Newtonian potentials. Analyses of 130 elliptical galaxy lenses find no evidence for a significant departure from general relativity at the 10–20% level on kpc–Mpc scales, with Yukawa-type screening scales only weakly constrained in the $$S[g,\mu,\lambda,\Psi,\gamma] = S_m[g,\Psi] + \frac{1}{16\pi} \int d^4x\,\sqrt{-g}\left\{ \frac{R-2\Lambda(\mu)}{G(\mu)} + \sum_p \lambda_p [\mu_p - f_p(g,\Psi,\gamma)] \right\}$$2 kpc–$$S[g,\mu,\lambda,\Psi,\gamma] = S_m[g,\Psi] + \frac{1}{16\pi} \int d^4x\,\sqrt{-g}\left\{ \frac{R-2\Lambda(\mu)}{G(\mu)} + \sum_p \lambda_p [\mu_p - f_p(g,\Psi,\gamma)] \right\}$$3 Mpc window (Guerrini et al., 2023).

In scalar-tensor chameleon-type models, effective $$S[g,\mu,\lambda,\Psi,\gamma] = S_m[g,\Psi] + \frac{1}{16\pi} \int d^4x\,\sqrt{-g}\left\{ \frac{R-2\Lambda(\mu)}{G(\mu)} + \sum_p \lambda_p [\mu_p - f_p(g,\Psi,\gamma)] \right\}$$4 allows successful fits to cluster and galaxy dynamics without dark matter for specific ranges of coupling $$S[g,\mu,\lambda,\Psi,\gamma] = S_m[g,\Psi] + \frac{1}{16\pi} \int d^4x\,\sqrt{-g}\left\{ \frac{R-2\Lambda(\mu)}{G(\mu)} + \sum_p \lambda_p [\mu_p - f_p(g,\Psi,\gamma)] \right\}$$5 and interaction length $$S[g,\mu,\lambda,\Psi,\gamma] = S_m[g,\Psi] + \frac{1}{16\pi} \int d^4x\,\sqrt{-g}\left\{ \frac{R-2\Lambda(\mu)}{G(\mu)} + \sum_p \lambda_p [\mu_p - f_p(g,\Psi,\gamma)] \right\}$$6. Constraints require $$S[g,\mu,\lambda,\Psi,\gamma] = S_m[g,\Psi] + \frac{1}{16\pi} \int d^4x\,\sqrt{-g}\left\{ \frac{R-2\Lambda(\mu)}{G(\mu)} + \sum_p \lambda_p [\mu_p - f_p(g,\Psi,\gamma)] \right\}$$7–$$S[g,\mu,\lambda,\Psi,\gamma] = S_m[g,\Psi] + \frac{1}{16\pi} \int d^4x\,\sqrt{-g}\left\{ \frac{R-2\Lambda(\mu)}{G(\mu)} + \sum_p \lambda_p [\mu_p - f_p(g,\Psi,\gamma)] \right\}$$8 on kpc–Mpc scales while preserving Solar System tests, suggesting a density- (and thus scale-) dependent screening (Mota et al., 2011).

5. Scale-Dependence, Gravitational Slip, and Structure Formation

Scale-dependence in $$S[g,\mu,\lambda,\Psi,\gamma] = S_m[g,\Psi] + \frac{1}{16\pi} \int d^4x\,\sqrt{-g}\left\{ \frac{R-2\Lambda(\mu)}{G(\mu)} + \sum_p \lambda_p [\mu_p - f_p(g,\Psi,\gamma)] \right\}$$9 generates observable consequences in cosmic structure formation primarily through the linear growth rate and gravitational slip. In conformal Newtonian gauge, the gravitational slip parameter $\lambda_p$0 is sourced by the difference in scale-dependent potentials; the magnitude of RG corrections to slip is typically subdominant compared to the impact on the growth index $\lambda_p$1 (Hamber et al., 2011). The growth rate observable $\lambda_p$2 is modified by $\lambda_p$3, introducing scale and time dependence that can potentially alleviate tensions such as the $\lambda_p$4 discrepancy between RSD and CMB datasets (Zhumabek et al., 2024). However, present analyses indicate that any $\lambda_p$5 at cosmological scales is tightly limited to $\lambda_p$6–$\lambda_p$7, with only a mild capacity to shift concordance parameters.

6. Future Directions and Experimental Prospects

Advances in survey precision, increased constraining power of joint weak lensing and RSD datasets, and improvements in scale-setting methodology (covariant or otherwise) will continue to test the smallest allowed departures from GR, constraining the RG parameter space and possible new fundamental mass scales down to $\lambda_p$8 Mpc (Baker et al., 2014). New theoretical developments in quantum gravity, effective field theory beta-functions, and higher-derivative terms may further clarify the allowed forms and scales of running. Laboratory tests of $\lambda_p$9, improved Hubble constant measurements, and high-resolution strong lens modeling can probe the scale-dependence of $\gamma_{\alpha\beta}$0 on smaller scales and over larger dynamic ranges.

In summary, the scale-dependent effective gravitational constant represents a central interface of quantum gravity, cosmology, and gravitational phenomenology. While current observations impose stringent limits on its variation over cosmic and local scales, the concept constitutes a robust and systematically improvable framework for testing the interplay between gravity’s low-energy limit and possible new physical scales (Hipólito-Ricaldi et al., 2024, Baker et al., 2014, Bertini et al., 6 Aug 2025, Bertini, 2024, Zhumabek et al., 2024).