What would happen if we had a crystal structure but only gravitational interactions? The idea is simple. Let's say we arrange similar bodies (call them planets, ions, anything) in an infinite crystal structure, but the only possible interactions are gravitational interactions.
A sensible guess is, the system will be unstable, it won't change unless it's perturbed, and if it is, it would not go back to the previous state after the perturbation.
Now, that was an obvious thing to guess. My question is, what else would change? What properties would disappear? Can we recover some of them?
Could we have, for example, lattice vibrations, and so on?  
 A: Much of the behavior of a crystal is governed by thermodynamics.
The thermodynamics of purely gravitational systems are quite counterintuitive. For one example, the heat capacity of a gravitational system is negative: when you add energy, it gets cooler. (There's a proof of this in Schroeder's thermo textbook, but if you know some astronomy you already know one consequence of this: if the energy output of a star's core increases because of a change from hydrogen fusion to helium fusion, the star gets brighter but cooler. We call these "red giants.") So if you have in mind "crystals do X," you have some hard thinking to do about why X happens and whether a gravitational system would do the same sort of thing.
What makes crystals stable is an interplay between attractive and repulsive forces. In gravitational systems you have only attraction. A disordered gravitational system eventually collapses to become a star-forming region. An ordered gravitational system, without any restoring repulsive forces, would rapidly become disordered, and from there eventually collapse to become a star-forming region.
If you consider an infinite gravitationally-bound uniform-density system, you should recover big-bang cosmology.
