Spherical space in the Newtonian limit: The cosmological constant

Abstract

We compute the cosmological constant of a spherical space in the limit of weak gravity. To this end we use a duality developed by the present authors in a previous work. This duality allows one to treat the Newtonian cosmological fluid as the probability fluid of a single particle in nonrelativistic quantum mechanics. We apply this duality to the case when the spacetime manifold on which this quantum mechanics is defined is given by R×S 3 . Here R stands for the time axis and S 3 is a 3-dimensional sphere endowed with the standard round metric. A quantum operator Λ satisfying all the requirements of a cosmological constant is identified, and the matrix representing Λ within the Hilbert space L 2 S 3 of quantum states is obtained. Numerical values for the expectation value of the operator Λ in certain quantum states are obtained, that are in good agreement with the experimentally measured cosmological constant.

Hyperbolic space in the Newtonian limit: The cosmological constant

“…Actually the real Universe is flat but has Λ > 0. The same ambiguity in the sign of Λ, due to an undetermined dimensionless factor, already arose in our analyses of flat space [8] and of spherical space [19].…”

“…As already done in refs. [8,19] we will restrict our attention to radially symmetric states, i.e., to l = 0, m = 0. Let us denote the ψ k00 (r) more simply by ψ k (r).…”

“…[33,34], as well as the holographic bound, have been exploited in refs. [6,8,10,19]. Altogether we see that a lot of useful information can be extracted via this indirect approach, where one tries to probe the fabric of the cosmos by means of simpler, or possibly better known, physical systems.…”

“…[8], and to the Einstein Universe in ref. [19]. These toy Universes are respectively described by the manifolds R×R 3 and R×S 3 .…”

“…In this paper we extend the analysis of refs. [8,19] to the case when the spatial manifold is H 3 , 3-dimensional hyperbolic space. Our Newtonian Universe is thus R × H 3 , with the real line standing for the time axis.…”