Post-Newtonian expansion of scale-dependent gravity

Technical Summary: Post-Newtonian Expansion of Scale-Dependent Gravity

Problem Statement
General Relativity (GR) remains the standard theory for gravitational interactions, successfully explaining Solar System data and predicting phenomena like gravitational waves. However, theoretical and phenomenological challenges persist, including the nature of the dark sector, cosmological parameter tensions, singularities, and the non-renormalizability of gravity at the quantum level. Renormalization Group (RG) approaches suggest that gravitational couplings, specifically Newton's constant $G$ and the cosmological constant $\Lambda$, may vary with the energy scale $\mu$ due to remnant low-energy quantum effects. A critical challenge in "scale-dependent gravity" is the "scale-setting procedure"—the covariant identification of the renormalization scale $\mu$ with physical quantities. Previous works established a covariant scale-setting procedure consistent with conservation laws, but the specific implications of this framework for Solar System tests via the Parameterized Post-Newtonian (PPN) formalism had not been fully explored.

Methodology
The authors apply the full Will–Nordtvedt PPN formalism to a scale-dependent gravity framework where $G$ and $\Lambda$ are treated as running couplings dependent on a scalar scale $\mu$.

1. Field Equations: The study utilizes modified Einstein equations where $G$ and $\Lambda$ are functions of $\mu$. The matter source is modeled as a perfect fluid.
2. Scale-Setting: The authors adopt the covariant identification proposed in previous work [25], where the renormalization scale is defined by the scalar quantity $\mu^2 \equiv \frac{1}{3}u^\alpha u^\beta G_{\alpha\beta}$. This choice is motivated by its ability to reproduce standard RG relations (e.g., $\mu \propto 1/r$ for static sources) and its compatibility with energy-momentum conservation ($\nabla_\alpha T^{\alpha\beta} = 0$).
3. Consistency Relations: To ensure conservation of the energy-momentum tensor, a consistency relation between the running of $G$ and $\Lambda$ is derived. The authors assume $G^{-1}$ can be expanded as a power series in a scalar field $\phi = \mu^2$, and $\rho_\Lambda$ is determined by integrating the consistency relation.
4. Post-Newtonian Expansion: The metric is expanded around Minkowski spacetime in powers of $v/c$. The authors expand the field equations up to the first post-Newtonian (1PN) order, determining the metric components $h_{00}$, $h_{0j}$, and $h_{ij}$ to the requisite orders ($O(v^4)$, $O(v^3)$, and $O(v^2)$ respectively).

Key Contributions and Results

• Metric Recovery: The analysis demonstrates that at the Newtonian order and the 1PN order, the metric components $h_{00}$, $h_{0j}$, and $h_{ij}$ coincide exactly with those of standard General Relativity. The standard PPN parameters (such as $\gamma$ and $\beta$) retain their GR values.
• Emergence of a New Potential: A deviation from standard GR appears only in the $O(v^4)$ term of the $g_{00}$ component. This deviation manifests as a new gravitational potential, denoted as $T$, defined by the Poisson equation $\nabla^2 T = -4\pi \rho^2$. This potential is proportional to the square of the mass density.
• Physical Implications of the New Potential: The authors find that this new potential modifies the definitions of pressure and internal energy within the fluid but does not affect the center-of-mass orbits of self-gravitating bodies at the 1PN order.
• Comparison with Previous RG Studies: The paper contrasts these results with previous Solar System analyses of scale-dependent gravity (e.g., Refs. [28, 32]). In those works, deviations from GR were significant enough to constrain model parameters. The authors attribute the absence of such constraints in their results to two factors:
  1. The specific covariant scale-setting procedure used ($\mu^2 \propto u^\alpha u^\beta G_{\alpha\beta}$), which differs from scale identifications based on the Newtonian potential or Runge-Lenz vector dynamics used in prior studies.
  2. The absence of derivatives of $G$ in their field equations, which suppresses the post-Newtonian deviations reported in earlier analyses.

Significance and Claims
The paper claims that the scale-dependent gravity framework, when implemented with the specific covariant scale-setting procedure and conservation consistency, is consistent with current high-precision Solar System tests. The fact that the theory reproduces standard GR predictions for the PPN parameters is presented as a "positive and nontrivial result," reinforcing the viability of the scale-dependent approach as a valid extension of GR.

The authors emphasize that while the theory passes Solar System bounds, the new potential $T$ suggests that deviations from GR may become observable at higher orders or in different contexts. They suggest that investigating higher-order corrections (e.g., 3PN order) and the emission of gravitational waves by binary systems are promising avenues for future detection of RG effects, as these regimes may be sensitive to the internal structure of compact bodies where the new potential plays a role. The work does not propose new experimental setups but highlights the necessity of extending the PPN formalism to higher orders to test the theory further.