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The diatomic molecule containing one ordinary hydrogen nucleus and one deuterium nucleus is an isotopic form of molecular hydrogen. It has a weak electric dipole and can supply low-temperature molecular line cooling in primordial gas when its abundance is sufficient.

The dark-matter halo population follows gravitational hierarchical galaxy formation: many low-mass dark-matter haloes and an exponentially rare high-mass tail. The Press-Schechter halo mass function captures this broad form. A constant conversion from dark-matter halo mass to luminosity would simply rescale that distribution and predict too much light from both very small and very large systems. The luminosity function instead has a shallower faint end and a sharp bright-end decline, often described by a Schechter function,Define the halo star-formation efficiency as ${ϵ}_{\ast }(M_h)=M_{\ast }/(f_b{M}_h)$, distinguishing this integrated baryon conversion from the star-formation efficiency per free-fall time. If the stellar mass-to-light ratio is ${{\rm Y}}_{\ast }$, a simple one-central-galaxy mapping is $L={ϵ}_{\ast }{f}_b{M}_h/{{\rm Y}}_{\ast }$. For a monotone mapping without scatter,Thus a mass-dependent conversion efficiency and its Jacobian change the shape, not just the horizontal scale, of the luminosity function. Scatter, satellites and the stellar mass-to-light ratio add further differences.

In shallow potential wells, photoheating can prevent gas accretion, while stellar feedback from radiation, stellar winds and supernovae heats or ejects gas. These effects lower halo star-formation efficiency toward low mass, helping flatten the faint end. The need for atomic or molecular line cooling also affects the smallest systems. Intermediate-mass dark-matter haloes retain gas and cool efficiently, so the integrated conversion reaches a maximum around galactic dark-matter halo masses.

In massive dark-matter haloes, long optically thin gas cooling times and stable hot atmospheres reduce fresh cold-gas supply. Active-galactic-nucleus feedback can replenish the thermal energy lost in cooling and suppress a cooling flow, reducing halo star-formation efficiency and sharpening the bright cutoff. Massive group or cluster dark-matter haloes contain many distinct galaxies rather than one object with luminosity proportional to their entire mass. Galaxy mergers redistribute stars and can grow already formed bright galaxies, but do not turn all hot dark-matter halo gas into new stars. The luminosity function is the gravitational dark-matter halo population filtered through strongly mass-dependent baryonic physics. The plotted abundance and efficiency curves are qualitative examples, not observational fits or calibrated mass-to-light relations.

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 $\mathrm{erg}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$; 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.

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 $T\gtrsim 10^6$–$10^7\mathrm{K}$, 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. The atomic threshold is a cooling-channel limitation, not a universal minimum gas temperature.