Non-Minimally Coupled Dark Matter

• Non-minimally coupled dark matter refers to models where dark matter interacts directly with spacetime curvature, modifying gravitational dynamics beyond standard general relativity.
• The framework introduces coupling functions in the action that alter Einstein's equations via terms like the Ricci scalar or tensor, impacting rotation curves and density profiles.
• These models offer a potential solution to the core-cusp problem by incorporating effective pressure and density-dependent interactions that mimic MOND-like behavior.

Non-Minimally Coupled Dark Matter

Non-minimally coupled dark matter (NMC-DM) refers to theoretical models in which the dark matter sector interacts directly with spacetime curvature beyond the minimal coupling prescribed by general relativity (GR). In these frameworks, the stress–energy tensor of dark matter sources gravity via non-trivial, typically density-dependent or field-dependent, coupling functions—leading to modifications of the gravitational field equations and, consequently, to the dynamics of dark matter halos, cosmological evolution, and potential observable signatures distinct from standard cold dark matter (CDM) models.

1. Theoretical Formulation and Types of Non-Minimal Coupling

The starting point for NMC-DM is the action functional, in which additional interaction terms couple dark matter to curvature invariants. Unlike minimal coupling—where the matter Lagrangian interacts only via the metric—NMC terms can couple dark matter variables directly to geometric quantities, most notably the Ricci scalar $R$ or the Ricci tensor $R_{\mu\nu}$. The general action takes the form

$$S = \frac{c^3}{16\pi G_N} \int d^4x \sqrt{-g} \left\{ [1 + \psi(\rho)] R + R_{\mu\nu} \xi(\rho) u^\mu u^\nu \right\} + S_{\rm DM} ,$$

where

• $\rho$ is the dark matter mass density,
• $u^\mu$ the four-velocity,
• $\psi(\rho)$ and $\xi(\rho)$ are dimensionless coupling functions, and
• $S_{\rm DM}$ is the canonical dark matter action.

These functions may depend on additional parameters such as temperature, such that the non-minimal coupling "switches on" only in regions of high density or low temperature (e.g., galactic centers), ensuring that standard GR is recovered elsewhere (e.g., in the Solar System) (1108.1728). Extensions generalize the functional dependence to scalar fields or other geometric invariants, enabling connections to scalar–tensor or vector–tensor frameworks and even to Horndeski-type theories.

2. Modified Field Equations and Newtonian Limit

Variation of the non-minimally coupled action with respect to the metric yields modified Einstein equations, which, coincident with the choices of $\psi$ and $\xi$, introduce new terms involving spatial gradients and Laplacians of these coupling functions and the dark matter density:

$$G^{\mu\nu} = \frac{8\pi G_N}{c^2} \rho u^\mu u^\nu + \text{[derivative and curvature corrections involving } \psi(\rho), \xi(\rho)] .$$

A notable property is that for the functional forms considered (i.e., avoiding higher than second derivatives), the equations remain of second order in the metric, thus evading the introduction of Ostrogradsky ghosts or other instabilities.

In the Newtonian (weak-field, non-relativistic) limit, the modified Einstein equations reduce to a Poisson-type equation for the gravitational potential $\Phi_N$:

$$\nabla^2 \Phi_N = 4\pi G_N \left[ \rho - \frac{\rho_{\rm fid}}{2} \nabla^2\tilde{\xi}(\rho) + \rho_{\rm fid} \nabla^2\tilde{\psi}(\rho) \right] ,$$

where $\rho_{\rm fid}$ sets the scale for the non-minimal effects. The presence of Laplacian terms (i.e., spatial derivatives of the density or coupling functions) implies that gravity is sourced not only by the local mass density but also by its inhomogeneities.

This mechanism directly impacts galactic-scale gravitational potentials, potentially flattening density cusps or modifying rotation curves independently of baryonic feedback, and can, with suitable tuning, mimic phenomenology akin to Modified Newtonian Dynamics (MOND) (1108.1728).

3. Fluid Properties and Internal Dark Matter Dynamics

Non-minimal coupling alters the effective stress–energy tensor of the dark matter fluid, introducing both isotropic and anisotropic components in addition to the usual pressureless dust term. The induced isotropic correction functions as an effective pressure, which may address the core–cusp discrepancy by providing additional support against gravitational collapse at small radii. The anisotropic corrections can induce net rotation or even non-trivial velocity anisotropy in the dark halo, impacting phase-space structure and caustic formation.

In scenarios motivated by Bose–Einstein condensation (BEC) of dark matter, the onset of condensation (associated with a sufficiently low temperature and high density to trigger macroscopic coherence) corresponds to the emergence of a new spatial scale—the healing length. When this scale is comparable to the local curvature radius, gradient terms analogous to the quantum potential

$$V_Q = -\frac{\hbar^2}{2m} \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}}$$

appear in the effective fluid equations, matching the form of the non-minimal coupling-induced Laplacian corrections. Thus, condensation phases naturally provide an origin for density-dependent NMC (1108.1728).

4. Phenomenological Consequences and Comparison with CDM

The implications of NMC-DM extend to both galactic and possibly larger scales:

• Galactic Rotation Curves: The modified Poisson equation allows for the derivation of rotation curves that can mimic observed baryon–DM scaling relations (e.g., Tully–Fisher law) without requiring fine-tuned baryonic feedback or new particle interactions. The predicted relation between baryonic and dynamical mass can approach that observed in late-type galaxies.
• Central Density Cores: The effective pressure term, even in the simple Newtonian limit, can stabilize the central potential against collapse, producing flat-core profiles consistent with cored halo observations, in contrast to the singular NFW or Einasto profiles typical of collisionless CDM-only simulations.
• MOND Phenomenology: By encoding a non-local contribution to the gravitational potential sourced by spatial density gradients, the models can reproduce features similar to MOND in the galactic regime, yet recover standard CDM behavior in the limit of vanishing NMC functions.

In all cases, the requirement $\psi(0) = \xi(0) = 0$ ensures compatibility with solar system and high-redshift cosmological constraints, as non-minimal effects are negligible for low DM density or high temperature.

Comparison Aspect	Standard CDM	Non-Minimally Coupled DM
DM gravitational clutch	Density only	Density plus spatial inhomogeneity
Core–cusp issue	Requires baryonic feedback	NMC-induced effective pressure
Rotation curve scaling	Derived from NFW/Einasto	Potential to realize scaling laws
Behavior at all scales	Single regime	NMC "switches on" in halos

5. Microphysical and Coarse-Grained Origins

The emergence of NMC in the dark sector may be justified by:

• Coarse-Grained Averaging: For a fluid of heavy, collisionless particles, the scale associated with coarse graining (the minimal scale above which a fluid description is valid) can match, or exceed, the local curvature scale, allowing DM to "sense" geometric gradients and rendering a non-minimal interaction in the effective field theory. However, the same logic could apply to baryons unless distinctive DM microphysics are imposed.
• Bose–Einstein Condensation: For bosonic dark matter candidates, condensation leads to the emergence of a healing length controlling long-range coherence, which couples naturally to the local curvature, especially as the system becomes degenerate. This process is temperature and density dependent and may dynamically select where and when non-minimal corrections become relevant.

This suggests non-minimal coupling is not an ad hoc assumption but may arise as an effective description in the collective, low-temperature regime of certain DM candidates.

6. Global Physical Consistency and Theoretical Constraints

The second-order nature of the field equations avoids ghost instabilities common to higher-derivative extensions. Parameterizations for the coupling functions can be constructed such that the theory is consistent with tested predictions of GR in controlled regimes, and strong constraints from the solar system, cosmological background expansion, and linear perturbation theory can be satisfied if $\psi(\rho),\xi(\rho)$ are sufficiently suppressed at low densities.

No new degrees of freedom are introduced into the gravitational sector provided the derivative structure does not exceed second order, distinguishing this class from higher-derivative or higher-curvature modifications.

7. Broader Implications and Future Directions

The NMC paradigm provides a unifying geometric mechanism capable of interpolating between the phenomenological successes of CDM at cosmological scales and MOND-like behavior at galactic scales. By embedding DM microphysics (e.g., condensation) or fluid-scale features into the gravitational sector, it offers a dynamical and potentially falsifiable alternative to both particle and modification-of-gravity frameworks.

Key open directions include:

• Determining the precise density and/or temperature-dependence of $\psi(\rho),\xi(\rho)$ most compatible with cosmic structure formation and kinematic data across all scales.
• Constraining the parameter space with high-resolution rotation curve data, strong and weak lensing measurements, and local probes of gravity.
• Investigating potential observational signatures (e.g., in the cosmic microwave background, gravitational lensing, or halo substructure) that distinguish NMC-DM from both standard CDM and modified gravity without DM.

A plausible implication is that if dark matter condensation or coarse graining at galactic scales is realizable, non-minimal coupling terms may be a robust, testable consequence in the broader context of dark sector physics (1108.1728).