Comparison of the gradient calculation and a power-law fit in log-log space. Left: N_eff(H)−n_H  relationship for the fiducial model (black), as shown in Fig. 2, and the polynomial fit for the dense gas (red). Right: Derivative, d log N_eff/d log n, as a function of density (black). The power-law fit is shown with the horizontal line. The dense gas regime with n_H > 10³ is highlighted with a bold line. The power-law relationship is annotated at the top.