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 $\Lambda^{1/4}$.
Highlights
Due to its major astrophysical and theoretical importance, the Buchdahl limit has been extensively investigated
We study the condensation process using the classical method of Jeans instability [39], generalized to arbitrary space–time dimensions, and by taking into account the possibility of the presence of viscous dissipative effects in the dark energy fluid
The Buchdahl inequality (30) is valid for all r ∈ [0, R] inside the star. This allows us to determine the upper bound on the mass–radius ratio, which is a natural consequence of the theory of general relativity, and to determine the more subtle lower bound on the mass–radius ratio when a nonzero cosmological constant is included in the analysis
Summary
Due to its major astrophysical and theoretical importance, the Buchdahl limit has been extensively investigated. In [12] it was shown that, in the framework of classical general relativity, the presence of a positive cosmological constant implies the existence of a minimum mass and a minimum density in nature These results follow rigorously from the generalized Buchdahl inequalities for D = 4, with > 0. The possibility that dark energy is not exactly a cosmological constant cannot be rejected a priori Taking into account this possibility, we obtain the Buchdahl and minimum mass limits in arbitrary space–time dimensions for dark energy obeying a linear barotropic equation of state. The more general case of a D-dimensional sphere of matter embedded in a dark energy fluid obeying an arbitrary barotropic equation of state is considered, and the corresponding limiting masses are derived. In estimating critical length and mass scales throughout this paper we use the approximate values c ≈ 2.998 × 1010 cm s−1, G ≈ 6.674 × 10−8 cm g−1 s−2, h ≈ 2π × 1.055 × 10−27 erg s, and ≈ 3 × 10−56 cm−2 for the fundamental constants
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