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Diffuse gas in photoionization equilibrium is often approximately described by the displayed temperature-density contrast relation. Photoheating and expansion cooling produce its thermal history. It is an empirical model for low-density gas, not the adiabatic equation of state of every intergalactic phase. Lyman-alpha forest line widths constrain both the temperature at a reference density and the slope through their lower envelope.

For Gaussian initial conditions and the Einstein-de Sitter second-order standard perturbation theory density kernel, the unsmoothed density contrast obeysThere are three choices of the second-order field and two connected Wick contractions; the angular average of $F_2$ is $5/7+(2/7)/3=17/21$. Thus the reduced skewness is $6(17/21)=34/7$. Smoothing changes this result, and the unsmoothed calculation requires a finite variance.