Scale-Dependent Gravity: Theory & Implications

• Scale-dependent gravity is a framework where constants like Newton’s G and Λ vary with scale, linking quantum corrections to classical general relativity.
• It employs covariant scale-setting and renormalization group methods to modify cosmological dynamics and alleviate tensions such as the H₀ discrepancy.
• The theory yields testable predictions for structure formation, black hole interiors, and solar system phenomena, bridging quantum field theory and gravitational physics.

Scale-dependent gravity is a class of theories in which the fundamental gravitational coupling parameters—most notably, Newton’s constant G and often the cosmological constant Λ—are promoted from universal constants to functions that vary with an energy, length, or curvature scale. Motivated by quantum field-theoretic considerations, renormalization group arguments, and phenomenological challenges to general relativity (GR) on astrophysical and cosmological scales, these frameworks aim to encapsulate the potential impact of quantum or semiclassical corrections on gravitational dynamics, while smoothly connecting with classical GR in the appropriate limits. Key arenas where scale dependence has been systematically analyzed include cosmology, astrophysics, compact objects, gravitational wave phenomenology, and experimental tests at Solar System and galactic scales.

1. Formal Foundations and Covariant Scale-Setting

The fundamental departure from classical GR in scale-dependent gravity is the replacement of fixed couplings with scale-dependent functions. The effective field equations take the schematic form

$$G_{\mu\nu} = 8\pi G(\mu)\left[ T_{\mu\nu} + g_{\mu\nu} \rho_{\Lambda}(\mu) \right]\ ,$$

with $G(\mu)$ and $\rho_\Lambda(\mu)$ functions of a renormalization scale $\mu$. Determining the physical meaning of $\mu$—the “scale-setting problem”—necessitates a covariant criterion. A recent prescription utilizes contractions of the Einstein tensor with the fluid four-velocity: $$\mu^2 = \frac{1}{3} u^\alpha u^\beta G_{\alpha\beta}$$ (Bertini, 12 Aug 2024, Bertini et al., 6 Aug 2025). This ensures scale identification is coordinate-independent and physically interpretable: in cosmology, for instance, it smoothly becomes proportional to $H^2$ for a Friedmann–Lemaître–Robertson–Walker background.

The requirement of energy-momentum conservation, enforced via the Bianchi identities, yields a consistency condition on the running forms: $$\frac{d\rho_\Lambda}{d\mu} + \frac{\rho + \rho_\Lambda}{G} \frac{dG}{d\mu} = 0,$$ which tightly constrains viable RG-induced running forms of $G(\mu)$ and $\rho_\Lambda(\mu)$ (Bertini, 12 Aug 2024). Expansions in $\mu$ or its quadratic variant $\phi = \mu^2$ are often adopted: $$G^{-1} = G_0^{-1} + \sum_n \nu_n \phi^n, \quad \rho_\Lambda = \rho_{\Lambda,0} + \frac{3}{8\pi} \sum_n \frac{n \nu_n}{n+1}\phi^{n+1}$$ (Bertini et al., 6 Aug 2025), ensuring compatibility with conservation laws and seamless matching to classical GR at low energies.

2. Cosmological Dynamics and Observational Tensions

Within cosmology, scale-dependent gravity models modify both the background expansion and the evolution of inhomogeneities. For example, the improved Friedmann equations derived from an RG-improved action can be cast as

$$\left(H^2 - H \frac{\dot{g}}{g}\right)/H_0^2 = \lambda(t) + \frac{\Omega_m}{a^3 g} + \frac{\Omega_r}{a^4 g}$$

with $g(t) = G(t)/G_0$ and $\lambda(t) = \Lambda(t)/\Lambda_0$ (Alvarez et al., 2020). Mild time variation in $g(t)$ introduces corrections that are designed to be negligible at early times (satisfying CMB and nucleosynthesis bounds) but become dynamically relevant at late times. Numerical and analytic studies demonstrate that such corrections can raise the late-time $H_0$ with minimal impact on early-universe physics, thus offering an elegant mechanism for alleviating the “$H_0$ tension” between local and high-redshift determinations (Alvarez et al., 2020, Bertini, 12 Aug 2024). Similarly, a bounded upper value for the Hubble rate $H_{\text{max}}$ (in models with quartic running of $G$ and $\rho_\Lambda$) can avoid the Big Bang singularity, yielding nonsingular cosmologies (Bertini, 12 Aug 2024).

Parameter fitting using CMB, supernovae, BAO, and RSD data constrains the running parameters (e.g., $\nu$ in $G_0 G^{-1}(W) = 1+\nu W$) to be exceedingly small $|\nu| \lesssim 10^{-5}$ (Hipólito-Ricaldi et al., 18 Nov 2024), resulting in only minor broadening of cosmological parameter uncertainties and preserving the successes of $\Lambda$CDM at the background and perturbative levels. While earlier results indicated possible relief of $\sigma_8$ tension via weaker gravity at $z\sim1$, comprehensive analyses with extended datasets show that the advantage becomes negligible (Panotopoulos et al., 2021, Hipólito-Ricaldi et al., 18 Nov 2024).

3. Scale-Dependent Structure Formation and Linear Growth

A distinctive consequence of scale dependence is the modification of the linear and nonlinear growth of cosmic structures:

• In scalar-tensor or $f(R)$ models, the growth rate $f_g(k, z)$ acquires explicit scale and redshift dependence. For Hu-Sawicki $f(R)$ gravity,

$$f_g(k,z; |f_{R}^0|) = \Omega_m(z)^\gamma \left\{1 + \exp\left[ -\frac{\alpha(z)}{\mathcal{K}^{5.5}} \right] \right\},\quad \mathcal{K} = k \lambda_C,$$

with $\lambda_C$ the scalaron Compton wavelength (Mirzatuny et al., 2019). This achieves sub-percent precision in fitting numerical solutions across $10^{-4} \le k \le 5\,\text{Mpc}^{-1}$ and $0 \le z \le 3$.

• Modified gravity models (e.g., $f(R)$) introduce an additional scale-dependent enhancement to the gravitational source term in the growth equation: $$\ddot{D}_+(k,t) + 2H \dot{D}_+(k,t) - \frac{3}{2}\Omega_m(a) H^2 \left[ 1 + \frac{2\beta^2 k^2}{k^2 + k^2_{\text{MG}}(a)} \right] D_+(k,t) = 0,\quad 2\beta^2 = 1/3,$$ with $k_{\text{MG}}(a) = a m_\phi(a)$, the modified gravity scale (Aviles, 24 Sep 2024).
• In the Newtonian gauge, corrections to the Poisson equation arise: $$\frac{k^2}{a^2}\phi = -\frac{1}{2F}\rho \delta \cdot \frac{g(F, U, \xi)}{1+\xi g(F, U, \xi)h(F, U, \xi)},$$ where $\xi(a, k) = 3a^2 H(a)^2/k^2$, capturing the key scale-dependent relativistic corrections on large subhorizon scales (Sanchez et al., 2010).

Validation against full numerical and N-body solutions shows that these analytic and semi-analytic approximations track the true growth remarkably well, especially for $k \gtrsim 0.01\,h\,\mathrm{Mpc}^{-1}$, while the standard scale-independent approach can err by 10% or more on large scales (Sanchez et al., 2010, Winther et al., 2017, Mirzatuny et al., 2019).

4. Implications for Astrophysics and Compact Objects

Scale-dependent gravity has been structurally explored in a variety of astrophysical contexts:

• In spherically symmetric black hole solutions, running Newton couplings $G(r)$ lead to modifications in the metric which, when matched to quantum effective potentials (e.g., Donoghue’s correction), yield nonsingular or finite-sized core (Planck star) geometries. For example,

$$G(r) = \frac{G_0 r^3}{r^3 + \omega G_0\hbar (r + \gamma G_0 M)}$$

with $\omega$ fixed by requiring consistency with the leading $1/r^3$ quantum corrections (Scardigli et al., 2022). The resulting lapse function $F(r)$ interpolates between Schwarzschild at large $r$ and a regular or hard-core interior at small $r$.

• The dynamical evolution of gravitational collapse in scale-dependent (or asymptotically safe) gravity shows that the fate of the collapse (regular halt, singularity at finite radius, etc.) depends critically on the sign and magnitude of the scale-dependent correction parameter $\omega$ (Hassannejad et al., 21 Oct 2024). For $\omega>0$, one obtains a regular nonsingular black hole; for $\omega<0$, a curvature singularity can occur at a finite radius. The matter equation-of-state and pressure profiles are accordingly modified.
• In relativistic star interiors, employing a scale-dependent $G(r)$ alters the structure equations. For constant or decreasing $G(r)$ with radius, strange quark stars can be slightly more massive and compact than in standard GR; crucially, the solutions remain stable and satisfy all energy conditions (Panotopoulos et al., 2021).
• The post-Newtonian expansion of scale-dependent gravity introduces a new $1$PN potential, $\mathcal{T}$, defined via

$\nabla^2 \mathcal{T} = -4\pi \rho^2,$

which modifies the local definition of pressure and internal energy, but does not impact center-of-mass orbital dynamics or observable PPN parameters, thus evading Solar System constraints (Bertini et al., 6 Aug 2025).

5. Experimental, Observational, and Phenomenological Constraints

Scale-dependent gravity models are subjected to a wide suite of experimental and observational tests:

• Solar System measurements, including light deflection and Shapiro time delay, are sensitive to scale-dependent corrections if they affect the post-Newtonian sector. However, since leading modifications in the 1PN metric can be absorbed into redefinitions of internal quantities rather than orbital dynamics, current tests do not constrain the leading terms in covariant scale-dependent PPN expansions (Jaekel et al., 2011, Bertini et al., 6 Aug 2025, Panotopoulos et al., 2021).
• Light deflection by the sun constrains scale-dependent parameters (e.g., the running parameter $\xi$ in the Schwarzschild–de Sitter solution) to be $\xi \leq 2.6 \times 10^{-12}$, suppressing any deviations from classical predictions in the weak-field regime (Panotopoulos et al., 2021).
• Cosmological "full shape" power spectrum analyses are especially sensitive to scale-dependent growth. State-of-the-art analyses using BOSS DR12, along with BBN and Planck priors, set the benchmark constraint $|f_{R0}| < 5.89\times 10^{-6}$ (68% CL) in the Hu–Sawicki model, comparable to or better than previous limits (Aviles, 24 Sep 2024, Rodriguez-Meza et al., 2023). Projection studies indicate that future spectroscopic surveys are unlikely to surpass these limits due to limitations in applying perturbation theory reliably at smaller scales.

6. Theoretical Contexts and Model Taxonomy

The landscape of scale-dependent gravity includes a variety of theoretical motivations and models:

• Renormalization group gravity / asymptotically safe gravity: Running couplings arise from the flow of fundamental parameters under the RG, leading to ultraviolet fixed point behavior and modifications at both high and low energies (Hamber et al., 2011, Scardigli et al., 2022, Hipólito-Ricaldi et al., 18 Nov 2024, Bertini, 12 Aug 2024).
• Scalar-tensor and chameleon models: Density-dependent effective masses and couplings can explain screening mechanisms, match dark matter profiles in clusters and galaxies with visible matter only, and remain compatible with local tests (Mota et al., 2011).
• Superfluid vacuum/gravity and emergent phenomena: Multiple-scale induced gravitational potentials explain galactic rotation curves, interpolate between Newtonian, flat, and nonflat (power law) velocity regimes, and yield an asymptotic Friedmann-like metric at the largest scales (Zloshchastiev, 2020).
• Covariant scale-setting and conservation: Covariant determination of the RG scale, with consistency conditions ensuring energy-momentum conservation, underpins viable model construction and perturbative stability in both background and cosmological contexts (Bertini, 12 Aug 2024).
• Background-preserving RG modifications: Theories constructed to match GR at the background level, with scale dependence manifest only at linear perturbation order, reproduce the acoustic peak structure in the CMB and offer very limited parameter shifts, rendering them observationally nearly indistinguishable from $\Lambda$CDM (Hipólito-Ricaldi et al., 18 Nov 2024).

7. Open Problems and Future Directions

Scale-dependent gravity frameworks have spurred development in observational approaches, simulation pipelines, and theoretical construction:

• Efficient and accurate fitting functions for scale-dependent growth rates, such as those for the Hu–Sawicki $f(R)$ model, are key tools for precision cosmology (Mirzatuny et al., 2019, Rodriguez-Meza et al., 2023).
• Perturbation theory extensions and simulation algorithms (e.g., fkPT, COLA with scale-dependent kernels and screening) have enabled MCMC exploration and robust confrontation with data (Winther et al., 2017, Rodriguez-Meza et al., 2023).
• The assessment of multi-messenger and large-volume surveys, and the integration of weak lensing, redshift space distortions, CMB, and gravitational wave observations, is ongoing and may enhance or further constrain scale-dependent effects (Aviles, 24 Sep 2024).
• Theoretical generalization—including additional degrees of freedom, higher-order terms in scale expansions, or more general RG flows—remains a frontier for addressing residual cosmological tensions and providing a concrete bridge between quantum field theory and gravitational phenomenology (Hipólito-Ricaldi et al., 18 Nov 2024, Bertini et al., 6 Aug 2025).

In summary, scale-dependent gravity offers a broad and systematic framework for investigating the running of gravitational couplings, potentially connecting quantum corrections with astrophysical and cosmological phenomena. Analytic and numerical results exhibit strong agreement with high-precision observational data in most regimes, while particular parameterizations remain under continued scrutiny for resolving fundamental tensions in current cosmology and for elucidating the ultimate quantum nature of gravity.