Spherical space in the Newtonian limit: The cosmological constant

TL;DR

This paper computes the cosmological constant for a spherical space in the weak gravity limit using a duality linking Newtonian cosmology to quantum mechanics, resulting in values consistent with observations.

Contribution

It introduces a novel duality approach that treats Newtonian cosmological fluids as quantum probability fluids on a spherical manifold, enabling direct computation of the cosmological constant.

Findings

Derived an operator for the cosmological constant in a spherical space.

Calculated expectation values that match experimental measurements.

Provided a quantum mechanical framework for cosmological constant estimation.

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 $\mathbb{R}\times\mathbb{S}^3$. Here $\mathbb{R}$ stands for the time axis and $\mathbb{S}^3$ is a 3-dimensional sphere endowed with the standard round metric. A quantum operator $\Lambda$ satisfying all the requirements of a cosmological constant is identified, and the matrix representing $\Lambda$ within the Hilbert space $L^2\left(\mathbb{S}^3\right)$ of quantum states is obtained. Numerical values for the expectation value of the operator $\Lambda$ in certain quantum states are obtained, that are in good agreement with the experimentally measured cosmological constant.