Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 73 / 3 / iv / Solution

Use high-resolution quasar spectra to fit Voigt profiles to Lyman-alpha forest lines near $z=3$, where the transition appears near $121.6(1+3)\simeq 486.4\mathrm{nm}$. Each fit estimates the neutral-hydrogen column density and the velocity-width parameter $b$. The one-dimensional Maxwell-Boltzmann velocity distribution has variance $k_{B}T/m_{\mathrm{H}}$; consequently the thermal optical-depth profile is proportional to $\exp [-(v-v_0{)}^2/b^2]$, with the thermal cutoff of Lyman-alpha line widthsFor example, a purely thermal $b=20\mathrm{km}{\mathrm{s}}^{-1}$ corresponds to $T\simeq 2.4\times 10^4\mathrm{K}$.

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, $b^2=b_{\mathrm{th}}^2+b_{\mathrm{nonthermal}}^2+b_{\mathrm{inst}}^2$. 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 $b$–$N_{\mathrm{HI}}$ 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 Schaye and collaborators' temperature analysis.