Relevant timescales in our model. The black line shows the overall merger timescale, t_merg. The dash-dotted green line shows the gas capture timescale, t_damp, (the shaded green histogram shows t_damp for 1g BBHs only). The dotted navy line shoiws the Type I migration timescale, t_migr,  I, (the shaded navy histogram shows t_migr,  I for 1g BBHs only). The dashed pink line shows the delay timescale, t_del, between the pair-up and the merger (the shaded pink histogram shows t_del for 1g BBHs only). The light blue line shows the gas dynamical friction timescale, τ₀, related to BBH pairing (Qian et al. 2024).