On the cosmological constant of flat FLRW spacetime

TL;DR

This paper models the cosmological constant in a flat FLRW universe using a quantum Schrödinger field, linking quantum states to observed cosmological parameters and entropy bounds.

Contribution

It introduces a quantum operator for the cosmological constant and connects its expectation value to the universe's matter content and entropy.

Findings

The quantum operator's expectation value matches the observed cosmological constant.

Identifies the quantum state representing current matter content.

Calculates the gravitational Boltzmann entropy within holographic bounds.

Abstract

We consider an exponentially expanding, flat, Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe filled with a free Schroedinger field. The probability fluid of the latter is used to mimic the cosmological fluid (baryonic plus dark matter), thus providing the matter density and pressure terms in the corresponding Friedmann-Lemaitre equations. We first obtain the eigenfunctions of the Laplacian operator on flat FLRW space. A quantum operator qualifying as a cosmological constant is defined to act on the Schroedinger field. We then compute the matrix representing the cosmological constant in the basis of Laplacian eigenfunctions. For an estimate of the orders of magnitude involved it suffices to determine the expectation values of this operator. The expectation value that best fits the experimentally measured value of the cosmological constant allows us to identify the quantum state of the Schroedinger field that best represents the matter contents (baryonic and dark) of the current Universe. Finally, the operator inverse (modulo dimensional factors) to the one representing the cosmological constant provides a measure of the gravitational Boltzmann entropy of the Universe. We compute its matrix in the basis of Laplacian eigenfunctions and verify that the expectation values of this entropy operator comply with the upper bound set by the holographic principle.