On the Newtonian limit and cut‐off scales of isothermal dark matter haloes with cosmological constant

We examine isothermal dark matter haloes in hydrostatic equilibrium with a ‘Λ‐field’, or cosmological constant Λ=Ω Λ ρ crit c 2 , where Ω Λ ≃ 0.7 and ρ crit is the present value of the critical density with h ≃ 0.65 . Modelling cold dark matter (CDM) as a self‐gravitating Maxwell–Boltzmann gas, the Newtonian limit of general relativity yields equilibrium equations that are different from those arising by merely coupling an ‘isothermal sphere’ to the Λ‐field within a Newtonian framework. Using the conditions for the existence and stability of circular geodesic orbits, the numerical solutions of the equilibrium equations (Newtonian and Newtonian limit) show the existence of: (i) an ‘isothermal region’ (0 ≤ r < r 2 ) , where circular orbits are stable and all variables behave almost identically to those of an isothermal sphere; (ii) an ‘asymptotic region’ ( r > r 1 ) dominated by the Λ‐field, where the Newtonian potential oscillates and circular orbits only exist in disconnected patches of the domain of r ; (iii) a ‘transition region’ ( r 2 ≤ r < r 1 ) between (i) and (ii), where circular orbits exist but are unstable. We also find that no stable configuration can exist with central density, ρ c , smaller than 2Λ, hence any galactic haloes which virialized at z < 30 in a ΛCDM cosmological background must have central densities of ρ c > 0.008 M ⊙ pc −3 , in interesting agreement with rotation curve studies of dwarf galaxies. Because r 2 marks the largest radius of a stable circular orbit, it provides a characteristic boundary or ‘cut‐off’ maximal radius for isothermal spheres in equilibrium with a Λ‐field. For current estimates of ρ c and velocity dispersion of virialized galactic structures, this cut‐off scale ranges from 90 kpc for dwarf galaxies, up to 3 Mpc for large galaxies and 22 Mpc for clusters. In a purely Newtonian framework, these length‐scales are about 10 per cent smaller, although in either case r 2 is between five and seven times larger than the physical cut‐off scales of isothermal haloes, such as the virialization radius or the critical radius for the onset of Antonov instability. These results indicate that the effects of the Λ‐field can be safely ignored in studies of virialized structures, but could be significant in the study of structure formation models and the dynamics of superclusters still in the linear regime or of gravitational clustering at large scales ( r ≈ 30 Mpc) .