Observations of spiral galaxies reveal a puzzling anomaly: the rotational velocities of stars and gas in the outer regions do not follow the expected Keplerian decline predicted by Newtonian dynamics and visible mass distributions. Instead, these velocities remain nearly constant, suggesting the presence of an unseen mass component—dark matter. This discrepancy, first noted by Vera Rubin and Kent Ford in the 1970s, has since become one of the most compelling pieces of evidence for dark matter's existence.
The standard cosmological model, ΛCDM (Lambda Cold Dark Matter), posits that dark matter is non-baryonic, collisionless, and interacts only gravitationally. While successful on large scales, ΛCDM faces challenges in explaining certain small-scale galactic phenomena, such as:
Given these challenges, researchers have explored alternative models, including fluid dynamical approaches. The premise is that dark matter may exhibit collective behavior akin to a fluid, governed by equations such as the Navier-Stokes equations or modified versions thereof. Such models can capture emergent phenomena like turbulence, vortices, and density waves—features that might align with observed galactic structures.
The connection between fluid dynamics and dark matter can be formalized through mathematical analogs. For instance:
A simplified model treats dark matter as an ideal fluid governed by the Euler equations coupled with the Poisson equation for gravity:
\[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0, \] \[ \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} = -\nabla \Phi - \frac{1}{\rho} \nabla p, \] \[ \nabla^2 \Phi = 4\pi G \rho, \]
where \(\rho\) is density, \(\mathbf{v}\) is velocity, \(\Phi\) is gravitational potential, and \(p\) is pressure. This system can exhibit wave-like solutions that might mimic dark matter density perturbations.
Some proposals introduce additional terms to account for possible dark matter self-interaction or non-Newtonian effects. For example:
\[ \rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}_{\text{ext}}, \]
where \(\mu\) is a dynamic viscosity coefficient and \(\mathbf{f}_{\text{ext}}\) represents external forces. Such models remain speculative but provide a framework for numerical experimentation.
Traditional N-body simulations treat dark matter as discrete particles interacting gravitationally. Fluid dynamical simulations, however, discretize the dark matter continuum onto a grid or mesh, solving the governing equations numerically. Key differences include:
| Aspect | N-Body Simulations | Fluid Dynamical Simulations |
|---|---|---|
| Resolution | Limited by particle count | Limited by grid size |
| Computational Cost | High for large N | High for fine grids |
| Emergent Phenomena | Clumping, filamentation | Turbulence, shocks |
Recent work has attempted to fit the Milky Way's rotation curve using a fluid dark matter model with a low viscosity coefficient. Preliminary results suggest that a fluid component could reduce the need for ad hoc halo adjustments.
Modified Newtonian Dynamics (MOND) proposes altering gravity to explain rotation curves without dark matter. Fluid dark matter models offer an alternative: emergent gravitational effects from fluid behavior without modifying gravity itself.
Advancements in computational power and observational techniques may soon allow stringent tests of fluid dark matter models. Key areas include: