Self-gravitating quantum stars with a globally relevant Bohm potential

Abstract

The microphysics underlying non-baryonic dark matter remains unknown. I derive the two-species Schrödinger-Poisson-Yukawa system for spin-1/2 dark-sector fermion fields, $\ddot{I}ˆ$ (mass $m_1$) and $\ddot{I}‡$ (mass $m_2$), coupled through a scalar mediator of mass $m_{\ddot{I}}•$ via a universal Yukawa coupling, within an orbital-free density-functional framework with the Kirzhnits gradient coefficient $\hat{I}{»}_B=1/9$. A central result is that the Bohm potential, far from being negligible in the Thomas-Fermi regime, contributes a species-dependent surface-energy correction analogous to the nuclear liquid-drop model: the heavier fermion species generates an outward quantum-pressure wall whilst the lighter species provides an inward surface tension, with degeneracy pressure furnishing the bulk confinement. In the single-species Schrödinger-Poisson limit the ground state recovers the benchmarked invariants $M_{\dim }\simeq 3.883$ and $x_T\simeq 2.562$, yielding $M{R}_T\simeq 9.95\hat{I}{»}_B{\hslash }^2/(G{m}_1^2)$. For polytropic index ${\hat{I}}^3=5/3$ the mass-radius relation satisfies $R\propto M^{-1/3}$; for ${\hat{I}}^3=4/3$ a limiting mass emerges above which no stable equilibrium exists. Illustrative configurations span $M=10^{-8}$-$5M_{\odot }$, $m_1\sim 10^{-14}$-$10^{-6}eV$, and radii from a few~km to $\sim 10^3{R}_{\odot }$, with gravitational-wave contact frequencies in the Einstein Telescope and LISA bands and microlensing signatures accessible to current surveys. The predictive rigidity of the resulting mass-radius relation, in which the single microphysical parameter $m_1$ determines the equilibrium radius once the total mass is specified, furnishes a reproducible, first-principles reference for constraining the dark-fermion mass in multi-component dark sectors.

16 pages, 10 figures, accepted for publication in Physical Review D