Minimal Gravitational Interactions

• Minimal Gravitational Interactions are defined as gravitational theories employing the Einstein-Hilbert action with the least possible ingredients, ensuring minimal coupling to matter.
• They explain phenomena such as flat rotation curves and scaling laws in galaxies without invoking additional components like dark matter by relying on effective field theory approaches.
• This minimal framework also informs experimental setups and quantum gravitational formulations by isolating essential couplings, thereby ensuring theoretical consistency and practical testability.

Minimal gravitational interactions refer to gravitational dynamics or coupling schemes in which either (i) the gravitational sector is assumed to comprise only the least possible set of ingredients—such as the Einstein-Hilbert action minimally coupled to matter fields—or (ii) gravitational phenomena arise from a minimal, tightly constrained set of assumptions without introducing additional components such as dark matter or extra degrees of freedom. Research on this topic explores foundational formulations, phenomenological scaling relations, interactions in quantum field theory, and experimental tests, with the aim of identifying the lowest-content gravitational descriptions compatible with a broad range of observations and theoretical consistency.

1. Minimal Gravitational Interaction Schemes

Minimality in gravitational interactions can be understood along several lines:

• Minimal Coupling (General Relativity): The standard Einstein-Hilbert action with matter coupled covariantly to the metric, i.e., $$S = \int \mathrm{d}^4x \sqrt{-g}\,[\frac{M_P^2}{2}R + \mathcal{L}_\text{matter}]$$, is minimal in the sense of general covariance and the equivalence principle (Sloan, 2014).
• Effective Field Theory and Contact Terms: For scalar fields, minimal gravitational interaction corresponds to absence of non-minimal couplings of the form $F(\phi)R$. However, graviton exchange in the presence of any non-minimal (Planck-suppressed) terms induces local ("contact") operators such as $F(\phi) T_\mu^\mu/M_P^2$ and $F(\phi)\partial^2 F(\phi)/M_P^2$ (Hill et al., 2020, Hill, 2022, Ghilencea et al., 2022). Through a Weyl transformation, these non-minimal components can be removed, yielding the minimal Einstein frame.
• Minimal Input Theories (Quantum Gravity): "Minimum information quantum gravity" postulates only statistical assumptions, a macroscopic spacetime, and the least possible degrees of freedom, from which the full hierarchy of particle interactions is derived (Mandrin, 2014). Here, gravity is minimal in its microlocal assumptions, yet provides the entire scaffolding for structure formation and quantum interaction hierarchies.
• Modified Gravity with Minimal Extensions: In galactic-scale phenomenology, minimal gravitational interaction may refer to models where the gravitational law is altered only by tightly constrained, ad hoc parameters (e.g., introducing a small graviton mass or a universal acceleration scale) in lieu of invoking dark matter (Trippe, 2012).

2. Mathematical Structures and Scaling Laws

Explicit mathematical frameworks for minimal gravitational interactions include:

• Density and Mass Profiles on Galactic Scales:

$\rho(r) \propto M_0\,\beta\,r^{-2}$

with the total enclosed mass,

$$M_\text{tot} = M_0\,\left[1 + 8\pi\frac{a_0}{a_c}\right],$$

where $$a_0 \approx 4.3 \times 10^{-12}\,\mathrm{m}/\mathrm{s}^2$$ is a universal acceleration scale and $a_c$ is the centripetal acceleration. This predicts flat rotation curves and scaling relations in galaxies without additional dark mass (Trippe, 2012).

• BRST Cohomology Constraints for Higher-Spin Fermions: The only consistent cubic vertices for massless higher-spin fermions and gravity require higher-derivative (non-minimal) structures; strictly minimal 1-derivative couplings are excluded (Henneaux et al., 2013).
• Weyl (Conformal) Transformations and Contact Terms: A theory with a scalar non-minimal coupling,

$$S = \int \sqrt{-g}\left[\frac{1}{2}(M_P^2 + F(\phi))R + \mathcal{L}(\phi)\right],$$

induces via graviton exchange contact operators $F(\phi)T_\mu^\mu/M_P^2$ and $F(\phi)\partial^2 F(\phi)/M_P^2$ that are equivalent to performing the Weyl rescaling $$g_{\mu\nu} \to g'_{\mu\nu} = \Omega^2(\phi)g_{\mu\nu}$$ with $\Omega^2 = 1 + F(\phi)/M_P^2$, to isolate the minimal sector (Hill et al., 2020, Ghilencea et al., 2022).

• Minimal Information Quantum Gravity (MIQG): By imposing symmetry constraints and conservation laws, the induced Dirac structure,

$\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu},$

emerges from minimal gravitational boundary terms alone (Mandrin, 2014).

3. Applications and Phenomenological Implications

Minimal gravitational interactions provide explanations for diverse phenomena:

• Galactic Phenomenology: Flat rotation curves, Tully-Fisher and Faber-Jackson relations, the baryonic mass discrepancy–acceleration relation, and Renzo’s rule can be quantitatively reproduced without invoking dark matter, but instead by a tightly constrained modification of the gravitational mass profile with a minimal set of extra parameters (Trippe, 2012).
• Structure Formation in Particle Physics: Minimal gravitational inputs, using only boundary term-induced symmetries, yield the Standard Model multiplet structure and predict higher-level interactions that could become visible at high energies (Mandrin, 2014).
• Cosmological Attractors: In Robertson–Walker geometries, minimal coupling of matter to gravity introduces a rescaling freedom in the phase space (from arbitrary fiducial cell volume), leading to Hamiltonian dynamics where attractor behavior (e.g., inflationary trajectories) emerges from a volume-weighted Liouville measure (Sloan, 2014).
• Gravitational Wave Signals and Early Universe Dynamics: Minimal gravitational interactions—such as pure inflaton scattering in the absence of explicit couplings—generate distinctive stochastic gravitational wave backgrounds during reheating, sensitive to the inflaton potential’s power law (e.g., $V\sim\phi^k$), with spectral properties determined by gravitational self-interaction mechanisms alone (Choi et al., 6 Feb 2024). In scenarios with non-minimal Higgs-curvature coupling, tachyonic instabilities driven by $\xi H^\dagger H R$ during kination efficiently produce both reheating and a stochastic gravitational wave background, with observable features linked to the inflationary scale and Standard Model parameters (Laverda et al., 6 Feb 2025).
• Scalaron–Graviton Conversion: In minimal Maxwell $f(R)$ gravity, the quartic dependence of the scalaron-to-GW conversion probability on background magnetic field strength,

$$\mathcal{P}_{S\to \text{GW}} = \frac{\kappa^4 B^4 d^2}{16\omega^2},$$

enhances gravitational wave production under primordial or neutron star magnetic fields, yielding signals potentially observable by the next generation of gravitational wave detectors (Capozziello et al., 8 Jul 2025).

• Laboratory Tests at Small Accelerations: Experimental setups employing suspended resonators achieve sensitivity to gravitational accelerations as low as $5.4 \times 10^{-12} \,\mathrm{m}/\mathrm{s}^2$. Results are in quantitative agreement with Newton’s law down to these regimes and provide an important bridge to galactic-scale anomalies, supporting the validity of minimal (Newtonian) gravitational interactions for baryonic matter at low acceleration (Bartel et al., 31 Jul 2024).

4. Quantum Field Theory, Higher-Spin, and Renormalization Group

Foundational results indicate important constraints and subtleties in quantum field theoretic constructions:

• Exclusion of Strictly Minimal Higher-Spin Couplings: Consistent gravitational couplings of massless higher-spin fermions require vertices of higher derivative order and exclude the minimal one-derivative ("minimal") vertex in flat space (Henneaux et al., 2013).
• Contact Terms and Renormalization Group: Induced local operators from graviton exchange in scalar–tensor theories modify the effective potential and RG flow, showing that calculations performed in the Jordan frame ignoring these contact terms produce incorrect $\beta$-functions. The Einstein frame, reached through a Weyl transformation and absorbing these contact terms into redefined couplings, is the unique frame in which minimal gravitational interactions persist and physical results are unambiguous (Hill et al., 2020, Hill, 2022, Ghilencea et al., 2022).
• Frame (In)dependence: The RG running of couplings in scalar–tensor theories does not in general commute with the transition between Jordan and Einstein frames unless contact terms are properly accounted for, emphasizing the operational importance of the minimal frame for extracting physical predictions (Ghilencea et al., 2022).

5. Theoretical and Experimental Boundaries of Minimality

Limits and scope of minimal gravitational interactions are characterized by:

• Toy Model Nature and Ad Hoc Assumptions: For scaling laws that explain galactic rotation curves, the approach is phenomenologically successful but depends on ad hoc assumptions (e.g., a universal acceleration $a_0$, non-zero graviton mass, and non-self-interacting gravitons) (Trippe, 2012). The scheme fails at solar system and cosmological scales.
• Necessary Role of Gravity at Particle Scale: Soliton-like, horizon-free solutions for elementary particles only exist with the gravitational coupling included ($\gamma \equiv (m\sqrt{G}/e)^2 \neq 0$ in the Lagrangian); as $\gamma\to 0$ (neglecting gravity), such regular solutions disappear, pointing to an indispensable gravitational role in extended particle models (Alharthy et al., 2021).
• Quantum-to-Classical Crossover: In few-body Newtonian systems, minimal gravitational interaction produces rich and sometimes nonintuitive dynamics (e.g., finite-time singularities with four particles on a line escaping to infinity in finite time), illustrating the complexities and possible singularities even with the minimal classical rule set (Dhar, 2023).
• Observational Prospects: Signals from minimal gravitational interactions—such as those from neutral mode-resonant conversion under strong magnetic fields, or gravitational waves from minimal preheating—often reside at the edge of current or near-future experimental reach, informing detector design and providing sharp, distinctive spectral signatures (e.g., monochromatic or multiple-harmonic peaks) (Choi et al., 6 Feb 2024, Capozziello et al., 8 Jul 2025).

6. Summary Table: Key Minimal Gravitational Mechanisms

Minimal Scheme	Core Mechanism	Phenomenological/Experimental Impact
Modified Newtonian with $a_0$	Graviton mass, universal accel., scaling law	Flat rotation curves, Tully-Fisher, no dark matter
Scalar–tensor Einstein frame	Weyl transform, induced contact terms	RG flow, physical observables, frame consistency
Minimum information quantum gravity	Boundary symmetries, minimal statistical input	Derivation of SM structure, extended interaction hierarchy
No minimal higher-spin coupling	Exclusion of 1-derivative vertex by BRST cohomology	Only higher-derivative, non-minimal interactions
Laboratory Newtonian tests	Suspended resonator, $\sim 10^{-12}~\mathrm{m}/\mathrm{s}^2$ sensitivity	Supports Newton’s law at low accel. levels
Minimal GW production (reheating, magnetic fields)	Inflaton condensate, scalaron-magnetic conversion	Predictive GW spectra, links to inflation and magnetar parameters

7. Outlook and Open Directions

The paper of minimal gravitational interactions continues to influence gravitational theory, the search for alternatives to dark matter, quantum gravity foundations, precision experimental design, and cosmological model-building. Whether in the context of quantum field theory, effective field theory, gravitational wave astronomy, or laboratory tests of Newtonian gravity at low accelerations, minimality serves both as a guiding principle and a phenomenological constraint, delimiting the necessary ingredients for a sufficiently predictive gravitational theory. Persistent questions involve the extension of minimal frameworks to fundamentally relativistic, quantum, or strongly self-gravitating regimes, and the extent to which astrophysical or cosmological phenomena can be encompassed by minimal modifications alone.