Scale dependence of the effective gravitational constant from functional renormalization group

Technical Summary: Scale Dependence of the Effective Gravitational Constant from Functional Renormalization Group

Problem Statement
General Relativity (GR), while successful in explaining experimental data to date, faces theoretical challenges regarding singularities, suggesting it may not be the final theory of gravity. A potential avenue for extending GR is treating the gravitational constant ($G$) not as a fixed parameter, but as a scale-dependent coupling constant that varies with energy scale, analogous to interactions in Quantum Field Theory (QFT). Previous works have investigated the mutual effects between the gravitational constant and the cosmological constant. However, the specific influence of including other fundamental interactions, such as electromagnetism, on the running of the gravitational constant remains to be systematically explored. Furthermore, the constraints imposed by observational data on the parameters governing this scale dependence require rigorous examination.

Methodology
The authors employ the Functional Renormalization Group (FRG) method, specifically utilizing the Wetterich equation derived from a truncated effective average action. The study defines scale-dependent dimensionless quantities for the gravitational constant ($g_k \equiv k^2 G_k$), the cosmological constant ($\lambda_k \equiv k^{-2} \Lambda_k$), and the fine-structure constant ($\alpha_k$).

The investigation proceeds through a systematic comparison of renormalization group (RG) equations under different interaction scenarios:

1. Gravitational Interaction Only: Deriving the RG equation for $g_k$ using a specific regulator function $R^{(0)}(z) = z/(e^z - 1)$.
2. Gravity + Cosmological Constant: Including the running of $\lambda_k$.
3. Gravity + Electromagnetism: Including the running of $\alpha_k$.
4. Full Coupling: Considering gravity, electromagnetism, and the cosmological constant simultaneously.

The authors analyze the solutions to the resulting RG equations, focusing on the behavior of $g_k$ as a function of the scale parameter $t = \ln(k/k^*)$. They identify two undetermined constants, $B_1(0)$ and $B_2(0)$, which arise from the regulator function and cannot be fixed from first principles. The study maps the "advisable" regions of the $B_1(0)$-$B_2(0)$ parameter space where the gravitational constant remains well-defined across all physical scales (i.e., avoiding singularities or unphysical limits).

Finally, the theoretical scale dependence is confronted with existing observational data. The authors relate the energy scale $k$ to physical observables such as distance, time, spacetime curvature (Kretschmann scalar), and temperature, using constraints from:

• Gravitational wave event GW170817 (comparing source and observer scales).
• Cosmological observations (recombination epoch vs. current epoch).
• Lunar laser ranging (time variation of $G$).
• Solar system tests (Newtonian inverse-square law).

Key Contributions and Results

1. Independence of Interaction Type: A primary finding is that the form of the RG equation governing the scale dependence of the effective gravitational constant ($g_k$) is qualitatively independent of the inclusion of other interactions. Whether the system includes only gravity, gravity plus the cosmological constant, or gravity plus electromagnetism, the equation for $\partial_t g_k$ remains:
   $$\partial_t g_k = (2 + \eta_N) g_k$$
   where $\eta_N$ depends on $g_k$ and the constants $B_1(0)$ and $B_2(0)$. The authors conclude that extra interactions do not affect the running coupling constant of the more fundamental gravitational interaction in this framework.

2. Parameter Constraints for Physical Scales: The analysis reveals that for the gravitational constant to be defined across all physically accessible scales (avoiding a divergence at a specific limit scale), the parameters $B_1(0)$ and $B_2(0)$ must satisfy specific inequalities. These "advisable regions" are visualized in the $B_1$-$B_2$ plane. Specifically, if $B_2(0) \leq 0$, $B_1(0)$ can take any value; however, if $B_2(0) > 0$, strict bounds apply to $B_1(0)$.

3. Observational Exclusion of Generic Solutions: When comparing the theoretical solutions (with generic parameter choices from the advisable regions) against observational data, the authors find that the constraints are extremely stringent.

   • GW170817: The constraint on the effective $G$ difference between the source (km scale) and observer (40 Mpc scale) is limited to a relative difference ratio of $|\Delta G/G_N| \lesssim 0.2$. This implies that for the scale difference $e^t \approx 10^{21}$, the parameters must be tuned such that $G$ varies negligibly.
   • Cosmology: Constraints on $G$ at recombination versus today ($|\Delta G/G_N| \lesssim 0.05$) similarly rule out the generic parameter sets used in the paper's illustrative examples (Figs. 2–7).
   • Time Variation: Lunar laser ranging constraints on $\dot{G}/G$ ($< 10^{-14} \text{yr}^{-1}$) require $|\eta_N| \lesssim 10^{-65}$.

4. Final Parameter Bounds: The observational data effectively rule out the broad parameter spaces explored in the theoretical examples. The paper concludes that to satisfy observations, the parameters must lie in extremely narrow regions:

   • Either $B_2(0) < 0$ (with specific conditions), or
   • $|B_1(0)|$ must be vanishingly small (e.g., $|B_1(0)| \lesssim 10^{-42}$ for distance-based constraints, and even tighter for time-based constraints).
     Crucially, the paper does not rule out all solutions; rather, it identifies that the viable "advisable region" collapses to a narrow line centered around $B_1(0) \approx 0$.

Significance and Claims
The paper claims to systematically investigate the scale dependence of the effective gravitational constant using the FRG framework, explicitly testing the robustness of this dependence against the inclusion of electromagnetic and cosmological interactions. The authors assert that while the form of the RG equation is robust and independent of these additional interactions, the quantitative behavior is strictly governed by two undetermined parameters.

The significance of the work lies in its demonstration that observational data places severe constraints on the functional renormalization group description of gravity. The authors conclude that unless the regulator-dependent parameters ($B_1, B_2$) are tuned to extreme values (effectively making the scale dependence negligible at observable scales), the standard FRG approach with these specific regulator choices is inconsistent with current gravitational wave and cosmological observations. The work serves as a rigorous check, suggesting that any viable theory of scale-dependent gravity must predict a nearly constant $G$ across the vast range of scales probed by modern astronomy.