Window for the formation of BBH mergers as predicted by semi-analytical and detailed binary models. Colors mark the evolutionary outcomes of interacting BH+O-star systems across different initial periods and mass ratios, calculated with our semi-analytical rapid model of SMT evolution (left panel; Sect. 2.3) and compared with MESA binary models for two different assumptions on orbital shrinkage during mass transfer: fiducial (γ = γ_BH, center) and enhanced shrinkage due to L2 outflows (right). The initial BH mass is always M_BH; 1 = 10 M_⊙ and metallicity is Z = 0.1 Z_⊙ (see Fig. D.1 for other BH masses). Wide-non interacting systems are not shown. Solid blue lines mark systems in which the SMT evolution would lead to final orbital separation a_fin = 2 R_⊙ or 25 R_⊙, according to the semi-analytical model, which is approximately the range of separations needed for BBH mergers. The dashed red line is the boundary between stable and unstable mass transfer. The outcomes of the semi-analytical model are only applicable if the mass transfer is stable. In rapid codes, it is typically assumed to be related to a critical mass ratio (the vertical line at q_crit ≈ 0.28, left panel). In the MESA models, the stability is determined self-consistently at every timestep, and it is found to be entirely different. The mass transfer only remains stable in systems where the final separation is a_fin ≳ 10 R_⊙. This conclusion is robust against uncertainties of SMT evolution and holds even if enhanced orbital shrinkage is assumed (right), leading to BBH mergers with long delay times ≳1 Gyr (Sect. 3.4).