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.
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