Scale-Dependent Newton Coupling

• Scale-dependent Newton coupling is a concept where the gravitational constant varies with environment and scale, driven by a scalar field with nonminimal matter coupling.
• The Yukawa correction modifies the gravitational potential over a distance L, enhancing gravity at galactic and cluster scales without invoking particle dark matter.
• Astrophysical analyses using MCMC methods reveal scalar field parameters that mimic dark matter profiles, though some probes like SNeIa offer limited sensitivity.

Scale-dependent Newton coupling refers to the phenomenon, predicted or engineered in various theoretical extensions of general relativity, in which the gravitational coupling constant—usually identified with Newton’s constant, $G$—varies as a function of physical scale, energy, or local spacetime environment rather than being strictly universal. This concept arises in multiple physical contexts: scalar-tensor gravity with nonminimal couplings, quantum gravity scenarios (e.g., asymptotic safety), effective field-theoretic constructions with nonlocal operators, and as a phenomenological explanation for the galactic and cluster-scale anomalies commonly attributed to dark matter. The analysis summarized here centers on scale-dependent modifications in the Newton coupling induced by the interaction between a massive scalar field and ordinary matter, the resulting Yukawa corrections, and astrophysical implications including the potential to mimic dark matter phenomena (Mota et al., 2011).

1. Scalar Fields with Environment-Dependent Mass and Coupling

Scale-dependent gravitational coupling is realized through a scalar field $\varphi$ coupled nonminimally to visible matter, with both its mass and coupling to matter acquiring local, environment-driven dependence. The field is governed by an effective potential,

$$V_{\rm eff}(\varphi) = V(\varphi) + \sum_i \rho_i e^{\beta_i \varphi/M_{\rm Pl}}$$

where $V(\varphi)$ is the scalar potential, $\rho_i$ represent local matter densities, $\beta_i$ are coupling strengths, and $M_{\rm Pl}$ is the reduced Planck mass. The field’s equilibrium (minimum) is found from the stationary point

$$V_{,\varphi}(\varphi_{\rm min}) + \sum_i \frac{\beta_i}{M_{\rm Pl}} \rho_i e^{\beta_i \varphi/M_{\rm Pl}} = 0$$

and the mass squared for local excitations is

$$m^2 = V_{,\varphi\varphi}(\varphi_{\rm min}) + \sum_i \frac{\beta_i^2}{M_{\rm Pl}^2} \rho_i e^{\beta_i \varphi/M_{\rm Pl}}$$

As a result, $m$ increases with local density, making the scalar field “hide” (short-ranged) in high-density environments and “emerge” (long-ranged) in low-density or cosmological settings, affecting gravitational interactions over the corresponding scale $L \sim 1/m$.

2. Yukawa-Type Gravitational Coupling and Effective Potential

The emergence of a scale-dependent Newton coupling is manifest in the modified gravitational potential produced by a point mass in the presence of the scalar field: $$\psi(r) = -\frac{G}{r} [1 + 2\beta^2 e^{-mr}] = -\frac{G}{r} [1 + 2\beta^2 e^{-r/L}]$$ Here, $G$ is the standard Newton constant, $\beta$ is the dimensionless scalar-matter coupling, and $L=1/m$ is the scalar field's effective range. The form of the correction is a classic Yukawa potential, resulting in an augmented gravitational attraction at distances $r \lesssim L$. The scale $L$ itself is dynamically determined by the environmental matter content: in dense regions $L$ contracts, suppressing new gravitational effects, while at galactic and cluster scales $L$ can become large, leading to an effective enhancement (or scale-dependence) of $G$ on astrophysical scales.

3. Constraints from Astrophysical and Cosmological Systems

The feasibility and physical consequences of scale-dependent gravitational coupling are tested across three astrophysical systems:

• Type-Ia Supernovae (SNeIa): The variable coupling modifies the observed distance modulus by an extra logarithmic term,

$$\mu(z; \beta, L; \lambda) = 5 \log[(1+z) \int_0^z dz/h(z)] + \mu_0 + \frac{15}{4} \log\frac{G_{\rm eff}(z; \beta, L; \lambda)}{G_{\rm eff}(0; \beta, L; \lambda)}$$

However, this correction remains at the $<0.1\%$ level, making SNeIa a poor probe for scale-dependent Newton coupling.

• Low Surface Brightness (LSB) Galaxies: The circular velocity is determined from the full potential (Newtonian + Yukawa). The rotation curves can be matched to observations using only visible (stellar + gas) mass components, without invoking additional dark matter, provided the scalar field parameters are appropriately tuned.
• Galaxy Clusters: X-ray profiles yield baryonic mass and temperature, permitting the inference of a total mass profile under hydrostatic equilibrium. The scalar field–induced potential and its radial gradient can fully account for the discrepancy between the total and baryonic masses—the so-called dark matter profile—without the need for actual dark matter.

4. Interpretation as a Dark Matter Mimicker

The scalar field corrections, parameterized by $\beta$ and $L$, generate an effective “apparent” or “mimicked” dark matter profile via

$$M_{\rm dm,th}(r; \beta, L) \equiv \frac{r^2}{G} \frac{d\Psi_C}{dr}$$

where $\Psi_C$ is the Yukawa part of the potential. The shape and normalization of this effective mass profile matches the observed “missing mass” in both LSB galaxies and galaxy clusters. Thus, the model provides a gravitational, rather than particle-based, origin for phenomena usually ascribed to dark matter. The analysis also implies that a single scalar field with scale-dependent coupling can simultaneously explain cosmological (dark energy-like), galactic, and cluster-scale (dark matter-like) phenomena through its environment-dependent behavior.

5. Model Limitations and Observational Challenges

Despite success in fitting galaxy and cluster data, several limitations constrain the robustness of the scale-dependent Newton coupling hypothesis:

• The modification of the SNeIa luminosity distance is negligible, preventing tight constraints on $G_{\rm eff}(z)$ from supernova data.
• Cluster mass profiles are unreliable in the innermost regions ($r \lesssim 100$–150 kpc) due to the breakdown of hydrostatic equilibrium.
• LSB rotation curve fits are hampered by degeneracies (e.g., the stellar mass-to-light ratio), bi-modality in fits, and observational uncertainties in surface brightness profiles.
• Degeneracies in the Yukawa length scale $L$ and coupling $\beta$ with system properties necessitate high-quality, homogeneous, and extended datasets for meaningful parameter extraction and discrimination from standard dark matter scenarios.

6. Synthesis of Theoretical and Observational Results

The model is tested using statistical inference (MCMC techniques), yielding preferred values

• For LSB galaxies: $\beta \sim 0.11$–0.14, $L \sim 0.15$ Mpc (cosmological units).
• For clusters: $\beta \sim 2$–3, $L$ in the range of several hundred kpc.

Correlations between scalar field parameters and structural properties (e.g., rotation velocity, surface brightness) are found, indicating environmental feedback consistent with a scale-dependent scenario. In clusters, the effective mass profile arising from $\Psi_C$ agrees with the profile inferred from X-ray data. However, SNeIa remain largely insensitive to the running of Newton’s constant on relevant cosmic scales.

Taken together, these results lend support to the viability of a gravitating, massive scalar field with a scale-dependent coupling to matter as an alternative to particle dark matter and possibly dark energy. Yet, definitive confirmation awaits more precise data and systematic investigation of predicted correlations and environmental dependencies.

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In conclusion, a scale-dependent Newton coupling, implemented via a massive scalar field with environmentally sensitive mass and coupling (parametrized as a Yukawa correction to Newton’s law), offers a unified, gravitational explanation for the empirical success of dark matter and dark energy phenomenology at galactic and cluster scales. Its primary signature is a length-scale-dependent modification to gravitational interactions that can account for flat rotation curves and observed cluster masses without unseen matter, subject to the caveat that certain astrophysical regimes—most notably, SNeIa—provide only weak constraints due to the smallness of the effect (Mota et al., 2011).