Minimum initial density of the gas (ρ_g0, min), given in Eq. (C.7), required for a BBH with equal mass BH members so that each member reaches mass 30 M_⊙ by Bondi accretion inside this gas’ density environment and until time t_f when the gas is completely depleted, assuming a constant mass loss rate of the gas (C.1). For an exponential gas loss law (C.2), the time t₉₉ needed for 99% depletion is t₉₉ = 2.3t_f, for the same values of all other parameters. We also assume in both figures c_s = 5 km s⁻¹, σ = v_c = 6 km s⁻¹ where we denote, respectively, the speed of sound of the gas, the velocity dispersion of the stars and the velocity of the binary with respect to the cluster. We consider three cases, each member having initial mass m₀ = 5 M_⊙ (blue solid line), 10 M_⊙ (red dashed-dotted line) or 15 M_⊙ (yellow dashed line). Left panel: we assume an initially slightly hard binary with χ(0)=1 ⇔ a(0)=a_h(0). It is evident that in order for a BH with initial mass ∼(5 − 10) M_⊙ to grow up to 30 M_⊙ within t_f = 5 Myr, assuming by this time there is complete gas depletion, would require an initial gas’ density ∼(5 − 15)×10⁴ M_⊙ pc⁻³. These minimum density values correspond to the exponential gas loss law with 99% depletion at time t₉₉ = 11.5 Myr. The density values are half for t_f = 10 Myr for both linear and exponential cases. We took into account the hardening of the binary during accretion. Right panel: we calculate ρ_g0, min w.r.t. the initial hardness expressed by the quantity logχ⁻¹(0) for t_f = 5 Myr in the linear case of gas loss. This time corresponds to t₉₉ = 11.5 Myr in the case of exponential gas loss. It is evident that soft binaries accrete gas more effectively than very hard ones. A hard binary a(0)/a_h(0)=0.1 with BH members each of mass (5 − 10) M_⊙ requires ρ_g0, min = (8 − 50)×10⁴ M_⊙ pc⁻³ while a soft one a(0)/a_h(0)=10 requires only (5 − 13)×10⁴ M_⊙ pc⁻³. By the time the gas is depleted a soft binary gets hard in case of isotropic accretion as demonstrated below, in Fig. 2.