Top: Cooling time at constant entropy as a function of temperature, for ﹩S_{100}=1﹩. Since ﹩t_{\mathrm{cool}\,}\propto S^{3/ 2}_{100}﹩, changing ﹩S_{100}﹩ only shifts the curves up or down the vertical axis. The cooling time has a deep minimum at ﹩T_{\mathrm{min}\,}=2.3\times 10^{5}﹩ K. If the cooling time exceeds the Hubble time at T_min, the gas will never be able to cool. Bottom: Critical entropy ﹩S_{\mathrm{crit}\,}﹩ for which ﹩t_{\mathrm{cool}\,}( T_{\mathrm{min}\,}) > t_{\mathrm{H}\,}﹩, as a function of redshift.