Panel (a): Redshift evolution of the total gas cross section, κ _tot = κ _g ( R _g / R ₂₀₀) ², for the best-fit combined model. The black curves represent different halo masses. The key indicated log  M for each line type. The two gray curves are the toy model results, discussed in Section 6.2, for log  M = 11.0 and 10.5 (upper and lower curves, respectively). Panel (b): Same as panel (a), but results are shown for the 2σ upper limit on κ ₁ shown in Figure 5. Gray curves have been shifted up to compare with the halo model results.