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, $ψ$ (mass $m_1$) and $χ$ (mass $m_2$), coupled through a scalar mediator of mass $m_ϕ$ via a universal Yukawa coupling, within an orbital-free density-functional framework with the Kirzhnits gradient coefficient $λ_B=1/9$. 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 $M_{\mathrm{dim}}\simeq 3.883$ and $x_T\simeq 2.562$, yielding $M R_T\simeq 9.95\,λ_B\hbar^2/(G m_1^2)$. For polytropic index $γ=5/3$ the mass-radius relation satisfies $R\propto M^{-1/3}$; for $γ=4/3$ a limiting mass emerges above which no stable equilibrium exists. Illustrative configurations span $M=10^{-8}$-$5\, M_\odot$, $m_1\sim 10^{-14}$-$10^{-6}\, eV$, 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.