GRAVITATIONAL EQUILIBRIUM in THE PRESENCE of a POSITIVE COSMOLOGICAL CONSTANT

We reconsider the virial theorem in the presence of a positive cosmological constant Λ. Assuming steady state, we derive an inequality of the form ρ ≥ A(Λ/4πGN ) for the mean density ρ of the astrophysical object. The parameter A depends only on the shape of the object. With a minimum at Asphere = 2, its value can increase by several orders of magnitude as the shape of the object deviates from a spherically symmetric one. This, among others, indicates that flattened matter distributions like e.g. clusters or superclusters, with low density, cannot be in gravitational equilibrium. Due to its wide range of application as well as its generality, the virial theorem plays an important role in astrophysics. To derive the virial theorem, one requires only the collisionless Boltzmann equation. Assuming steady state, one of the important applications of the virial theorem is to deduce the mean density of astrophysical objects like galaxies, clusters and superclusters by observing velocities of a ‘testbody’ around them. It is clear that for conglomeration of matter, spread over large enough scales, the Hubble expansion of the universe will, in principle, oppose the gravitational equilibrium. It is also known that a positive cosmological constant Λ ≥ 0 accelerates this expansion (for a review on the cosmological constant and its problems, see [1]). If, as recent measurements seem to indicate [2], a positive cosmological constant enters the Einstein’s equations, the resulting space-time in the Newtonian limit (called Newton-Hooke space [3]) will inherit the expansion due to the Λ-term in the form of an external force. In this limit of the Einstein’s equations we can rederive the virial theorem and evaluate the conditions under which a steady state for a collection of matter is reached when, as it is the case here, we have two opposing forces: the attractive Newtonian force and the repulsive external Λ-force. In exploring the astrophysical significance of Λ > 0, we will make use of the revised virial theorem (other approaches to eventual astrophysical effects of Λ have been discussed in [4] and [5]). Before going into medias res of the virial theorem, we mention first some salient features of the Newtonian limit itself, including now the cosmological constant [4].