Evolution of BH masses m₁, m₂ (red lines) and BBH hardness (minus logarithm) χ(t)=a_h(t)/a(t) (blue lines) plotted until the time the gas reservoir of a primordial cluster is completely depleted for an exponential loss law (for a linear law the depletion time for the same mass growth is about half) so that the BHs attain the mass values of the GW signal GW150914, namely m₁(t_end)=36 M_⊙, m₂(t_end)=29 M_⊙. We assume c_s = 5 km s⁻¹, σ = 6 km s⁻¹ and the orbital velocity of the BBH w.r.t. the cluster to be constant v_c = 4.5 km s⁻¹ which corresponds to a circular orbit of the barycenter at a distance ∼0.15 pc from the center of the cluster. Assuming v_c = σ will add about 8 Myr. Left panels: we consider spherically symmetric Bondi accretion and solve the system (C.10) and (C.11). The initial gas density is ρ_g(0)=5 × 10⁴ M_⊙ pc⁻³. We find that the time needed is 12 Myr for a(0)/a_h(0)=1, corresponding to the soft-hard boundary, in the upper panel and m₁(0)=8.35 M_⊙, m₂(0)=8.03 M_⊙, while we consider a soft binary in the lower panel with a(0)/a_h(0)=5 ⇔ χ(0)=0.2 and m₁(0)=7.91 M_⊙, m₂(0)=7.54 M_⊙. The hard BBH gets about 62 harder, while the soft 74 times harder becoming very hard at the end of the process χ(t_end)=14.8. Both the mass increase and hardening effects are more intense for the soft binary in agreement with Fig. 2b. Right panels: we consider Eddington-limited accretion in a disk, as in Eq. (A.7), depicted with solid lines and we plot in the same graph the Bondi case with dotted lines. We assume initial gas density ρ_g(0)=10⁴ M_⊙ pc⁻³. The time needed for a similar growth to the Bondi case is about four times larger with respect to the left panels, namely 50 Myr with a little higher initial masses m₁(0)=11.85 M_⊙, m₂(0)=9.55 M_⊙. We note that in order to achieve the same final mass in the same time with Bondi accretion, the initial masses are much lower, namely m₁(0)=6.40 M_⊙, m₂(0)=6.29 M_⊙ in the hard case and m₁(0)=5.54 M_⊙, m₂(0)=5.40 M_⊙ in the soft one. In case of angular momentum preservation (or loss to the disk) the BBH gets harder in Eddington-limited accretion as well, though it is less severe than in the Bondi case. The soft binary gets hard attaining χ(t_end)=5.6.