Which hydrogen atoms can absorb the light?
Imagine a bright distant object as a lamp behind a thin cloud of hydrogen. The gas absorbs some colours more strongly than others. A dip in the light reaching a telescope is an absorption line. For Lyman-alpha absorption, an electron bound to a hydrogen atom takes up a photon and moves to a higher energy level.
Neutral hydrogen has an electron bound to its proton, with no net electric charge. A more energetic ultraviolet photon can remove the electron entirely. This is ionization. The remaining proton cannot produce the same Lyman-alpha absorption. A cloud can therefore become more transparent even when its total amount of hydrogen stays the same.
Electrons can be captured again, a process called recombination. In the dilute, pure-hydrogen comparison used here, photoionization and radiative recombination balance to determine the neutral fraction. At fixed temperature and total density, increasing the photoionization rate removes more of the atoms that could absorb.
Reading: Verner & Ferland, Atomic Data for Astrophysics: Radiative Recombination Rates, arXiv:astro-ph/9509083v1 · Independent hydrogen and sightline calculation materials
Heating changes the atoms' motions too
Temperature acts in two ways. Over the range considered here, from 5,000 to 30,000 kelvin, the radiative recombination rate falls as the gas gets hotter. With the ionizing light held fixed, fewer neutral atoms remain. That change in population already weakens the absorption.
The remaining atoms also move faster. Some move towards us along the line of sight; others move away. Their motions shift the colours at which they absorb, so the collection of atoms absorbs across a wider range. This is thermal Doppler broadening. For an isolated cloud, spreading the absorption can make its centre shallower even if the beam encounters the same amount of neutral hydrogen.
The thermal width grows with the square root of absolute temperature. Heating hydrogen from 10,000 to 30,000 kelvin multiplies the temperature by three, but increases this width by only the square root of three, about 1.73. Going from 5,000 to 10,000 kelvin gives a factor of about 1.41. These factors describe the thermal motion of one gas component; the width of several overlapping dips includes their different bulk velocities as well.
Reading: Verner & Ferland, Atomic Data for Astrophysics: Radiative Recombination Rates, arXiv:astro-ph/9509083v1 · Tepper García, Voigt profile fitting to quasar absorption lines, arXiv:astro-ph/0602124v1 · Independent hydrogen and sightline calculation materials
Two ways to let four fifths of the light through
A numerical example makes the ambiguity concrete. Keep one prescribed gas distribution, its large-scale velocities and the same sightline. At 10,000 kelvin and the default ionizing rate, the mean transmission across the selected spectral interval is about 70.2 percent. On average across that interval, roughly seven tenths of the background light gets through.
Double only the photoionization rate and the mean transmission rises to about 80.3 percent. Now restore the original rate and heat the gas to 30,000 kelvin instead. The mean transmission is about 80.4 percent. Almost the same mean can result from two different changes. The first leaves thermal motions unchanged; the second changes both recombination and thermal broadening.
Those percentages belong to this constructed sightline and finite spectral interval. They are not measurements of the universe's mean transmission. They show why reducing a spectrum to one transparency number loses useful information. The detailed shapes retain clues that the two averages have hidden.
Reading: Independent hydrogen and sightline calculation materials
A real spectrum carries a history
Bulk motion can broaden or blend absorption features. An instrument can blur a narrow feature too. Higher gas density also favours recombination. Interpreting a real spectrum means considering these effects alongside the random motions that temperature controls.
Earlier heating can leave another memory: pressure can spread out small density structures. Two regions with the same temperature today may have different density patterns because they were heated at different times. The numerical comparison above fixes the density and changes the present absorption physics; changing its temperature does not calculate a new expansion history. To use these spectra to study intergalactic temperature or dark matter, astronomers must also model gas structure and motion, and account for which kinds of absorption their observations can pick out.
Reading: Peeples et al., Pressure Support versus Thermal Broadening in the Lyman-alpha Forest, arXiv:0910.0256v3 · Independent hydrogen and sightline calculation materials