Is there actually a balance needed between pressure and collapse? Radiation pressure presumably doesn’t do anything to the constituent dm particles. Similarly, wouldn’t the particles in the star be on various elliptical trajectories and not collapse?
The DM particles would indeed be on random/chaotic orbits, unlike visible matter which can shed kinetic energy via EM interaction (collisions). But DM density near a mass concentration would still be higher than far away from anything massive.
Normal stars are in hydrostatic equilibrium, a density where the inward force exerted by gravity and the outward force exerted by the pressure of the hot plasma are balanced. In a dark star the situation would be similar, except the heat would be generated by DM annihilation rather than fusion (the heat from annihilation would keep the star too "puffy" to reach the core pressure and temperature required for fusion.
> Similarly, wouldn’t the particles in the star be on various elliptical trajectories and not collapse?
That's a common misunderstanding. Orbits around many bodies do not work that way, and the particles exchanging momentum so they collide or escape the cloud is normal.
You're forgetting about friction. A diffuse cloud is eventually going to become tighter and tighter as gravity draws it together and the individual orbiting particles within that cloud lose momentum from hitting each other.
Well, they say the DS would also have a decent amount of baryonic matter in it - some diffuse hydrogen and helium. If the pressure from the annihilations pushes the hydrogen and helium outwards, that could create some outward gravity to pull on the dark matter, right? Wait, except the outward gravitational pull inside of a hollow shell is zero because it all cancels.
Would there even be friction though? It sounds like the only interactions these hypothetical particles have is gravity and annihilation.
Yes, no friction for the DM particles. WIMP DM cannot collapse by shedding kinetic energy as heat like visible matter can. Thus the dark matter halos around galaxies. But gravity would still cause there to be a higher average density of DM inside and near the dark star (or indeed a modern-day galaxy) than far away from any massive objects.
> If the pressure from the annihilations pushes the hydrogen and helium outwards, that could create some outward gravity to pull on the dark matter, right?
But it's not on net being pushed outwards. It's in equilibrium. The outward push is on average exactly canceled out by the baryonic matter falling inwards due to friction. If it weren't, then everything would get either denser/sparser until equilibrium were regained. The point is that the forces are working in such a way as to maintain a stable equilibrium, like in normal stars.
