Equilibrium halo solutions of the Gross–Pitaevskii–Poisson system: the role of the particle number

Abstract

We investigate stationary halo-like solutions of the Gross–Pitaevskii–Poisson (GPP) system, which describes self-gravitating Bose–Einstein condensates with repulsive self-interactions, as a model for dark matter. We retain the boson mass

, the scattering length

$$a_s$$ a s

, and the total particle number N explicitly, treating the latter as an independent macroscopic control parameter of the equilibrium configuration. We solve the stationary GPP equations and explore a broad region of parameter space. The resulting equilibrium configurations are classified into ground-state, excited-state, and unbound branches according to their binding properties and nodal structure. We find that the ground-state branch occupies a well-defined region of the

$$(m_\phi ,N)$$ ( m ϕ , N )

plane whose location is highly sensitive to the self-interaction strength, whereas the excited-state and unbound regions exhibit a structure largely independent of the initial ansatz. By analyzing the converged solutions, we derive empirical scaling relations linking the characteristic halo radius

$$R_{99}$$ R 99

to the boson mass, scattering length, and total particle number. The recovered scaling reproduces the known mass–radius relation in the non-interacting limit, while finite self-interactions reveal an intermediate regime in which gravity, quantum pressure, and repulsive interactions jointly determine the equilibrium structure, deviating from the asymptotic Thomas–Fermi limit. As an astrophysical application, we show that ground-state solutions can reproduce the rotation curves of representative dwarf galaxies using only the solitonic component. Finally, we examine the impact of repulsive self-interactions on the location of the ground-state branch and discuss the implications for current Lyman-

$$\alpha $$ α

forest constraints. Although increasing

$$a_s$$ a s

shifts equilibrium solutions toward boson masses compatible with existing Lyman-

$$\alpha $$ α

bounds, the corresponding halo configurations fail to reproduce the observed dwarf-galaxy kinematics. Our results provide a systematic characterization of stationary GPP halos and establish a direct connection among the fundamental particle properties

$$(m_\phi , a_s)$$ ( m ϕ , a s )

, the macroscopic control parameter N , and observable galactic properties.