= Solution
Use high-resolution <quasar> spectra to fit <Voigt profiles> to <Lyman-alpha forest> lines near $z=3$, where the transition appears near $121.6(1+3)\simeq486.4\,\mathrm{nm}$. Each fit estimates the neutral-hydrogen <column density> and the velocity-width parameter $b$. The one-dimensional <Maxwell-Boltzmann velocity distribution> has variance $k_BT/m_{\rm H}$; consequently the thermal optical-depth profile is proportional to $\exp[-(v-v_0)^2/b^2]$, with the <thermal cutoff of Lyman-alpha line widths>
$$
\boxed{b_{\rm th}^2=\frac{2k_BT}{m_{\rm H}},
\qquad T=\frac{m_{\rm H}b_{\rm th}^2}{2k_B}.}
$$
For example, a purely thermal $b=20\,\mathrm{km\,s^{-1}}$ corresponds to $T\simeq2.4\times10^4\,\mathrm K$.
An observed broad line is not by itself a thermometer: <peculiar velocities>, bulk gradients, blending and instrumental resolution also broaden it. For independent Gaussian contributions, $b^2=b_{\rm th}^2+b_{\rm nonthermal}^2+b_{\rm inst}^2$. After modeling the instrumental response, an uncorrected total width gives an upper bound on the thermal <temperature>, not a lower bound. The narrowest lines at each <column density> are least affected by additional broadening. Their lower envelope in the $b$–$N_{\rm HI}$ plane constrains the <temperature-density relation of photoionized intergalactic gas>, often written $T=T_0(1+\delta)^{\gamma-1}$.
Calibrate the conversion between <column density>, density and width with hydrodynamic models incorporating the <cosmic ionizing background>, pressure smoothing, resolution and selection. It estimates $T_0$ and the slope rather than assigning one <temperature> to every absorber. Where aligned metal lines trace the same gas, comparing species of different atomic mass can separately estimate a common nonthermal contribution because thermal $b^2$ scales as $m^{-1}$. The forest lower-envelope method and its numerical calibration are developed in https://arxiv.org/abs/astro-ph/9906271[Schaye and collaborators' temperature analysis].
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