The self-gravitating gas in the presence of the cosmological constant

The non-relativistic self-gravitating gas in thermal equilibrium in the presence of a positive cosmological constant Λ is investigated. The cosmological constant introduces a force pushing outward all particles with strength proportional to their distance to the center of mass. We consider the statistical mechanics of the self-gravitating gas of N particles in a volume V at thermal equilibrium in the presence of Λ. It is shown that the thermodynamic limit exists and is described by the mean field equations provided N,V→∞ with N/V13 fixed andΛV23 fixed. That is, Λ→0 for N,V→∞. The case of Λ fixed and N,V→∞ is solved too. We solve numerically the mean field equation for spherical symmetry obtaining an isothermal sphere for Λ>0. The particle distribution becomes flatter when Λ grows compared with the Λ=0 case. Moreover, the particle density increases with the distance when the cosmological constant dominates. There is a bordering case with uniform density. The density contrast between the center and the boundary may be significantly reduced by the cosmological constant. In addition, the critical point associated to the collapse (Jeans') phase transition is pushed towards higher values of N/[TV13] by the presence of Λ>0.