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For isotropic specific intensity $J_{\nu }$, absorption cross-section ${\sigma }_{\nu }$, and threshold frequency ${\nu }_0$, the photoionization rate per target atom isFor a directed incident flux, replace $4\pi J_{\nu }$ by its spectral flux density. The factor $h\nu$ converts energy flux into photon flux. This rate alone does not imply photoionization equilibrium.

The quasar proximity effect is a reduction of nearby Lyman-alpha forest absorption caused by extra photoionization from the quasar. At fixed density and temperature in photoionization equilibrium, an added rate ${\Gamma }_Q$ changes the neutral fraction by the factor ${\Gamma }_{\mathrm{bg}}/({\Gamma }_{\mathrm{bg}}+{\Gamma }_Q)$. A host environment with positive density contrast can mask this reduction, and a variable or anisotropic source complicates its interpretation.

In photoionization equilibrium at fixed gas density and temperature, the extra quasar photoionization rate reduces neutral-hydrogen column density by the displayed factor. Its local rate is calculated from the quasar flux, while the isotropic background rate is ${\Gamma }_{\mathrm{bg}}=4\pi \int J_{\nu }{\sigma }_{\mathrm{HI}}(\nu )/(h\nu )d\nu$. Fitting the absorption suppression versus proper distance constrains the cosmic ionizing background. For an unperturbed column-density distribution proportional to $N^{-p}$ with $p>1$, the line abundance above a fixed threshold is reduced by $(1+\omega {)}^{1-p}$. Density enhancement, quasar age, anisotropy and redshift errors must be distinguished from this ionization effect.

The Lyman-alpha forest becomes denser and more opaque towards higher cosmological redshift, for a fixed absorption selection over the ordinary post-reionization range. The incidence is often parametrized as $d\mathcal{N}/dz=A(1+z{)}^{\gamma }$, with the index depending on the column density threshold and epoch. The evolution flattens at low redshift as the declining cosmic ionizing background partly offsets dilution of the intergalactic medium. For optically thin gas at fixed overdensity, photoionization equilibrium gives approximately $n_{\mathrm{HI}}\propto \alpha (T)(1+z{)}^6/\Gamma$, illustrating why density, temperature and ionizing radiation all matter.

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.