Evolution of the SFE and stellar mass in dark matter halos, tracing the halo growth with cosmic time. Left: Evolution of the SFE for three different halos of the same initial mass, M_h = 2 × 10¹¹ M_⊙, starting their evolution at z = 9.3, z = 6.0, and z = 2.25, respectively. The halo mass growth follows Dekel et al. (2013) parametrization, and the SFE is computed from the SHMR at the corresponding redshift and halo mass. The size of the points corresponds to the halo mass, also annotated above each point (in log scale). Right: Stellar mass growth in the same halos as the left panel, where M_⋆ = M_h ϵ_⋆ f_b. The envelope corresponds to the 1σ confidence interval propagated from the uncertainty in the SHMR. The dashed lines are computed in the same way, using the resulting SHMR from the FFB model at ${ϵ}_{\star }^{\mathrm{FFB}}=0.2$, while the transparent envelope encloses between no FFB and maximum efficiency $0.0<{ϵ}_{\star }^{\mathrm{FFB}}<1.0$ (Liu et al. 2024). This shows that halos of 10¹¹ M_⊙ starting their growth at z > 3.5 would be more efficient in assembling stellar mass by the time they reach 10¹² M_⊙.