A MONDest proposal: a fundamental model of Newtonian physics at cosmic scales

This talk presents a simple physical model of classical gravitation that qualitatively exhibits many of the characteristics of modern astronomical observations: 

* The laws of classical physics should be invariant under the group of translations of space and the group of rotations of space.
  * The force of gravity should be conservative. In a moderate or weak gravitational field, the Laplacian of the gravitational potential should be the matter content.  In particular, in a vacuum, the potential is harmonic.  (This rules out MOND, for example.)
  * In moderately large gravitational fields (e.g., solar systems), the physics should be approximately Newtonian and orbits Keplerian.
  * In weak gravitational fields (e.g., galactic halos), there should be a flattened velocity curve.

Moreover, there is a natural relativistic space-time variant of the model on a homogeneous space with positive cosmological constant (de Sitter space). The Newtonian background is a sub-Riemannian space modeled on the Heisenberg group. The cosmological solution is modeled on the spin group of de Sitter space time. In the Newtonian case, we show that the perturbative orbits the virial theorem of mechanics have a flattened rotation curve. The geodesics in the case of a cosmological vacuum are also considered. Other physical implications will be discussed as time permits.