Backward extrapolation of exponential closed-box star formation ($\Psi (t)=\Psi (t_0)e^{\alpha ϵ(t_0-t)}$)

A fixed-efficiency closed box with permanent locking fraction $\alpha$ predicts exponential decay of gas and gross star formation. Backward extrapolation therefore grows exponentially with lookback time. It requires a stated present age and assumes the same gas reservoir and efficiency over the entire interval. Cosmological inflow, outflow, mergers and changing gas conditions generally invalidate such an extrapolation over many depletion times.