On the newtonian limit and cut--off scales of isothermal dark matter halos with cosmological constant

Abstract

We examine isothermal dark matter halos in hydrostatic equilibrium with a cosmological constant Lambda =Omega_\Lambda rho_{crit}c^2, where Omega_\Lambda=0.7, and rho_{crit} is the present value of the critical density with h=0.65. The Newtonian limit of General Relativity yields equilibrium equations that are different from those arising by merely coupling an ``isothermal sphere'' to the Lambda-field within a Newtonian framework. The conditions for the existence and stability of circular geodesic orbits show the existence of (I) an ``isothermal region'' (0<r<r_2), 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 Lambda-field, where the Newtonian potential oscillates and circular orbits exist in disconnected patches of the domain of r; (III) a ``transition region'' (r_2<r<r_1), circular orbits exist but are unstable. We also find that no stable configuration exists with central density, rho_c, smaller than 2 Lambda, hence any galactic haloes which virialized at $z< 30$ in must have rho_c >0.008 M_\odot {pc}^3, in agreement with rotation curve studies of dwarf galaxies. Since r_2 marks the largest radius of a stable circular orbit, it provides a ``cut off'' radius. For current estimates of rho_c and velocity dispersion of galactic structures, this is around five times larger than the virialization radius. The effects of the Lambda$--field can hence be ignored in structure formation models, but could be significant in the dynamics of superclusters in the linear regime or of gravitational clustering at large scales.