The minimum mass of a spherically symmetric object in $D$-dimensions, and its implications for the mass hierarchy problem

TL;DR

This paper generalizes Buchdahl inequalities to arbitrary dimensions with a cosmological constant, exploring minimum mass and density limits, dark energy stability, and quantum implications across different space-time dimensions.

Contribution

It derives new mass bounds in higher dimensions with cosmological constant and analyzes dark energy stability and quantum mass scales.

Findings

Explicit minimum and maximum mass formulas for stable objects in D-dimensions.

Analysis of Jeans instability for dark energy with arbitrary equations of state.

Quantum mass scale linked to cosmological constant, suggesting dark energy as a quantum particle sea.

Abstract

The existence of both a minimum mass and a minimum density in nature, in the presence of a positive cosmological constant, is one of the most intriguing results in classical general relativity. These results follow rigorously from the Buchdahl inequalities in four dimensional de Sitter space. In this work, we obtain the generalized Buchdahl inequalities in arbitrary space-time dimensions with $\Lambda \neq 0$ and consider both the de Sitter and anti-de Sitter cases. The dependence on $D$, the number of space-time dimensions, of the minimum and maximum masses for stable spherical objects is explicitly obtained. The analysis is then extended to the case of dark energy satisfying an arbitrary linear barotropic equation of state. The Jeans instability of barotropic dark energy is also investigated, for arbitrary $D$, in the framework of a simple Newtonian model with and without viscous dissipation, and we determine the dispersion relation describing the dark energy$-$matter condensation process, along with estimates of the corresponding Jeans mass (and radius). Finally, the quantum mechanical implications of mass limits are investigated, and we show that the existence of a minimum mass scale naturally leads to a model in which dark energy is composed of a `sea' of quantum particles, each with an effective mass proportional to $\Lambda^{1/4}$.