Temperature-density relation of photoionized intergalactic gas ($T=T_0(1+\delta {)}^{\gamma -1}$)

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.