Dimensionless spin a(m) and a(t) for different values of initial spin: a ₀ = −1, 0, 1. The lower x-axis gives the mass accreted over the initial BH mass: when m/M ₀ = 1, the BH has accreted its entire initial mass. The upper x-axis gives spin evolution in megayears over an initial Eddington rate λ ₀. (Panel (a): the BH with an NT disk gains angular momentum and energy, and will spin up to a _eq = 1. (Panel (b): An MAD will spin down the BH to a _eq ≈ 0.07. The MAD spin-down timescale is very short (much shorter than the NT spin-up timescale): the BH reaches its equilibrium spin after accreting only half of its initial mass. For a fixed λ ₀, this spin-down timescale is independent of the initial BH mass. Dashed lines show BH spin evolution based on our GRMHD simulation results (computed using a spline interpolation of simulated data points, e _in(a) and l _in(M, a), shown in Figure 4) and solid thin lines show the spin evolution computed using the semi-analytic model described in Section 3.4: our model provides a good description of the simulation results.