= Solution
For an optically thin low-density gas in <collisional ionization equilibrium>, the <astrophysical cooling function> collects losses per pair of hydrogen nuclei into $\mathcal C=n_H^2\Lambda(T)$. Its units are $\mathrm{erg\,cm^3\,s^{-1}}$; the time unit is missing from the printed guide to the vertical-axis labels. For the <primordial atomic cooling curve>, only <hydrogen> and <helium> provide bound-state coolants.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-60-cooling.png]
{title=Schematic primordial cooling function with hydrogen and helium features, metal enrichment and photoionization effects}
Near $10^4\,\mathrm K$, <collisional excitation> of <hydrogen> becomes effective and subsequent photon emission removes thermal energy. The excitation rate is exponentially suppressed below the atomic energy threshold. <Atomic line cooling> and <collisional ionization> generate a strong hydrogen feature around a few $10^4\,\mathrm K$; helium excitation and ionization produce further structure around $10^5\,\mathrm K$. <Radiative recombination> also contributes. Once the gas is highly ionized, these bound-state losses weaken. At $T\gtrsim10^6$–$10^7\,\mathrm K$, <thermal bremsstrahlung> dominates, with approximately $\Lambda\propto T^{1/2}$ apart from slowly varying factors. The figure is an original qualitative sketch anchored to the two supplied values, not an atomic-rate calculation.
Increasing <galactic metallicity> adds many ionic transitions, producing <metal-line cooling> and substantially enhancing cooling, particularly in the $10^5$–$10^7\,\mathrm K$ range. Metals also provide low-energy fine-structure transitions below $10^4\,\mathrm K$. The change is not a uniform vertical shift: the positions and strengths of features depend on the elemental abundances and ionization state.
<Photoionization> removes bound electrons even where collisions alone would leave atoms neutral, often suppressing the hydrogen and helium line peaks and changing the metal-ion population. It also adds <photoionization heating>. The net thermal loss is then $\mathcal C-\mathcal H$, with possible heating-cooling equilibrium near $10^4\,\mathrm K$. A single density-independent $\Lambda(T)$ no longer describes every irradiated cloud: the answer also depends on radiation intensity, spectrum, shielding and density. The illustrative photoionized curve denotes altered cooling alone, not the net loss including heating.
Below $10^4\,\mathrm K$, primordial gas can use <molecular line cooling>, especially rotational and vibrational transitions of <molecular hydrogen> and, in suitable chemical conditions, <hydrogen deuteride>. Molecule formation and protection against photodissociation are essential. Enriched gas can additionally use <metal-line cooling>, other molecular species and <dust thermal emission>, with gas-dust energy exchange important at high density. At high <cosmological redshift>, residual free electrons can transfer energy to the <cosmic microwave background> through <Compton cooling by the cosmic microwave background> if the gas is hotter than the radiation. Expansion can also cool gas adiabatically, but neither that process nor <Compton cooling by the cosmic microwave background> is an $n_H^2$ atomic loss law. \b[The atomic threshold is a cooling-channel limitation, not a universal minimum gas temperature.]
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