On the total mass of closed universes with a positive cosmological constant

TL;DR

This paper extends the concept of total mass density to closed universes with a positive cosmological constant, establishing a lower bound and characterizing when this bound is achieved, especially in relation to de Sitter spacetime.

Contribution

It introduces a new lower bound for total mass density in universes with positive cosmological constant and characterizes the conditions for equality, extending previous notions to broader cosmological models.

Findings

Cosmological constant sets a sharp lower bound for total mass density.

Minimum total mass density occurs if and only if the universe is locally de Sitter.

The notion of total mass density is applicable to non-compact homogeneous cosmologies.

Abstract

The recently suggested notion of total mass density for closed universes is extended to closed universes with a positive cosmological constant. Assuming that the matter fields satisfy the dominant energy condition, it is shown that the cosmological constant provides a sharp lower bound for the total mass density, and that the total mass density takes this as its minimum value if and only if the spacetime is locally isometric with the de Sitter spacetime. This notion of total mass density is extensible to non-compact three-spaces of homogeneity of Bianchi class A cosmological spacetimes.