Self-interacting dark scalar spikes around black holes via relativistic Bondi accretion

We consider the spike mass density profile in a dark halo by self-consistently solving the relativistic Bondi accretion of dark matter onto a non-spining black hole of mass M. We assume that the dominant component of the dark matter in the halo is a Standard model gauge-singlet scalar. Its mass m ≃ 10^-5 eV and quartic self-coupling λ ≲ 10¹⁹ are constrained to be compatible with the properties of galactic dark halos. In the hydrodynamic limit, we find that the accretion rate is bounded from below, M _min = 96πG ² M ² m ⁴/λħ ³. Therefore, for M = 10⁶ M⊙ we have M ^min ≃ 1.41 × 10^-9 M _⊙ yr^-1, which is subdominant compared to the Eddington accretion of baryons. The spike density profile ρ ₀(r) within the self-gravitating regime cannot be fitted well by a single-power law but a double-power one. Despite that, we can fit ρ ₀(r) piecewise and find that ρ ₀(r) ∝ r ^-1.20 near the sound horizon, ρ ₀(r) ∝ r ^-1.00 towards the Bondi radius and ρ ₀(r) ∝ r ^-1.08 for the region in between. This contrasts with more cuspy ρ ₀(r) ∝ r ^-1.75 for dark matter with Coulomb-like self-interaction.