Black hole growth in non-rotating FFB clusters with a broad IMF. Shown is the cluster half-mass radius versus mass. The magenta curves and regions below them refer to clusters where core collapse occurs within 3 Myr, the lifetime of massive stars. Left: based on Eq. (4) (Portegies Zwart & McMillan 2002; Devecchi & Volonteri 2009), assuming either ⟨m⟩​ = ​10 M_⊙ (solid) or ⟨m⟩​ = ​100 M_⊙ (dashed). Right: based on Eq. (3) (Rizzuto et al. 2021, Eq. (4)), assuming m_max​ = ​100 (solid) or 300 M_⊙ (dashed). The blue and red lines refer to constant density n​ = ​10^3.5 cm⁻³ and surface density Σ​ = ​10^3.5 M_⊙ pc⁻² within the half-mass radius. Feedback-free starbursts are expected to occur near such densities. FFB clouds of the Jeans mass and radius, ∼10⁶ M_⊙ and ∼7 pc, lie above the core-collapse line, though not too far away if the IMF is very top-heavy, of a population-III.1 type, with ⟨m⟩​ ∼ ​100 M_⊙ (left) or m_max​ > ​300 M_⊙ (right). If the sub-Jeans clouds form by fragmentation with the same Σ, they should lie along the red line, such that core collapse will occur in 3 Myr for M_c ≤ 10⁵ M_⊙ and R_e ≤ 2 pc. The corresponding black-hole mass at the end of the FFB phase, Eq. (5), in addition to the initial mass of the massive star that became the VMS, is marked in the top axis, indicating M_bh​ ∼ ​10³ M_⊙ in ∼10⁵ M_⊙ clusters.