Constraints on the covering factor within R _g from the combined model. Inner and outer contours represent models with Δχ ² < 1 and 4 with respect to the best fit model, respectively. The dashed curve is the best-fit model at z = 0.6, while the dotted curve represents κ _g ( M) at z = 2.0. At z = 2, the decrease in κ _g above 10 ¹² h ⁻¹  M _☉ has little effect on the Mg ii statistics because the abundance of hot-mode halos is significantly attenuated at this and higher redshifts.