The cosmological constant of emergent spacetime in the Newtonian approximation

Abstract

We present a simple quantum-mechanical estimate of the cosmological constant of a Newtonian Universe. We first mimic the dynamics of a Newtonian spacetime by means of a nonrelativistic quantum mechanics for the matter contents of the Universe (baryonic and dark) within a fixed (i.e., nondynamical) Euclidean spacetime. Then we identify an operator that plays, on the matter states, a role analogous to that played by the cosmological constant. Finally we prove that there exists a quantum state for the matter fields, in which the above mentioned operator has an expectation value equal to the cosmological constant of the given Newtonian Universe.

Spherical space in the Newtonian limit: The cosmological constant

“…In this article we present a toy model of Newtonian Universe in positively curved space, in which a certain repulsive force plays a role analogous to that played by the cosmological constant Λ in general relativity. This follows previous work by the same authors [7], where we have computed the cosmological constant of flat, Newtonian space. In the present paper we extend this previous analysis to the case of positive (but constant) scalar curvature, still within the limit of weak gravity.…”

“…In this section we verify that the results obtained in S 3 correctly reduce to their counterparts in R 3 [5,7,9] as one lets R 0 → ∞.…”

“…Upon setting c 2 = 0, c 1 = 1/R 0 and expanding around χ = 0 we obtain V (χ) = 1/(R 0 χ), that is V (r) = 1/r, which is of course the Newtonian potential in flat space R 3 . The cosmological constant is represented quantum-mechanically as an operator that, in flat space, is proportional to the inverse of the squared radial distance ρ [7]. In R 3 , the standard choice is of course the centrifugal potential energy,…”

“…Newtonian cosmology succeeds in capturing some essential physics of the Universe (for a very incomplete sample see, e.g., refs. [1,2,4,5,7]). It also has the advantage that its mathematics is notably simpler than that of its general-relativistic parent theory.…”

“…We refer the reader to our paper [7] for a thorough explanation of our use of nonrelativistic quantum mechanics in order to model the Newtonian cosmological fluid. Since this feature is, to the best of our knowledge, new in the literature, let us reiterate: we will mimic the Newtonian cosmological fluid as the quantum mechanics of a single nonrelativistic particle within a given spacetime manifold.…”