Cosmology · The cosmic web, voids and intergalactic matter

Intergalactic gas and diffuse matter

Compare two mostly ionized hydrogen slabs with equal total hydrogen columns but different neutral columns under a supplied equilibrium model.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Distinguish total hydrogen, neutral hydrogen and free-electron columns.
Calculate a conditional ionization correction with density, radiation and recombination stated.
Explain why gas phases and thermal history complicate an intergalactic matter census.

Diffuse matter is not one uniform gas phase

Intergalactic gas occupies different density, temperature and ionization conditions. Radiation, expansion, cooling and shocks can affect those states, so a single tracer does not automatically count all of the gas between galaxies.

Neutral-hydrogen absorption can be sensitive to a small neutral part of a largely ionized hydrogen population. Converting that column into total hydrogen requires a physical ionization model.

Free electrons, protons, neutral atoms, helium and metals contribute differently to possible observations. A number column of one species is distinct from a total mass column, and both differ from a volume density.

Use two ideal hydrogen slabs to show this distinction quantitatively. The construction is not a complete inventory of intergalactic phases or a measurement of the cosmic baryon fraction.

The neutral fraction depends on density and radiation

Let x be the neutral fraction of pure hydrogen, so n_HI = xn_H and n_e = n_p = (1−x)n_H. Here n_H counts all hydrogen nuclei per physical volume, not only neutral atoms.

Supply a common external photoionization rate Γ per neutral atom and a common Case A recombination coefficient α_A. Assume ionization equilibrium, negligible collisional ionization and negligible attenuation of the ionizing background.

The balance is Γx = α_An_H(1−x)². Define η = α_An_H/Γ, giving x = η(1−x)². Its stable physical root is x = 2η/[1+2η+√(1+4η)].

The Case A prescription lets recombination photons escape in this optically thin account. Ionization equilibrium is assumed, not established by a computed relaxation history; it is neither Saha equilibrium nor proof of thermal or mechanical equilibrium.

Equal total columns can produce unequal neutral columns

Choose adjacent slabs along one sightline with common area and total proper path length L. Slab A has length 0.9L and density n_ref; slab B has length 0.1L and density 9n_ref.

Define N_ref = n_refL and supply α_An_ref/Γ = 0.001. Both slabs have total hydrogen column N_H = 0.9N_ref, while their η values are 0.001 and 0.009.

The physical neutral fractions are approximately 0.000998005 in A and 0.00884156 in B. Both slabs are mostly ionized, but the denser slab has a larger equilibrium neutral fraction under the same supplied radiation and coefficient.

Multiplying each fraction by its total hydrogen column gives N_HI/N_ref ≈ 0.000898204 and 0.007957400. Slab B contributes half of the total hydrogen column but about 89.86 percent of the neutral column.

Two supplied pure-hydrogen slabs at one epoch; lengths are proper and all columns are divided by N_ref = n_refL
Slab	Length/L	n_H/n_ref	η	Neutral fraction x	Total H column	Neutral H column	Electron column
A	0.9	1	0.001	0.000998005	0.9	0.000898204	0.899102
B	0.1	9	0.009	0.00884156	0.9	0.007957400	0.892043

A neutral fraction needs a stated weighting

The combined total hydrogen column is 1.8N_ref and the neutral column about 0.0088556045N_ref. Their ratio is a hydrogen-column-weighted neutral fraction about 0.00491978.

The electron column is about 1.7911444N_ref. In this pure-hydrogen model it equals the proton column, but it differs from the total hydrogen-nucleus column because neutral atoms contribute no free electron.

Column units are inverse area: physical number density in m⁻³ multiplied by proper path length in metres gives m⁻². These line integrals do not by themselves specify the volume occupied by that gas.

The absolute N_ref can be chosen small enough for the imposed optically thin ionizing-field assumptions. Only ratios are supplied, so no actual density, temperature, path length, atomic coefficient or measured column is inferred.

Two adjacent pure-hydrogen slabs share one sightline area and proper total length L. A occupies 0.9L with hydrogen-nucleus density n_ref; B occupies 0.1L with density 9n_ref. The top path blocks have exact drawing widths 432 and 48 pixels, proportional to those supplied physical length fractions, with the narrow block labelled B. Both total hydrogen columns are 0.9N_ref with N_ref = n_ref L. Assume a common prescribed external photoionization rate Γ per neutral atom and Case A recombination coefficient α_A, negligible collisional ionization and negligible attenuation. Case A lets recombination photons escape. Neutral fraction x solves Γx = α_A n_H(1−x)², with n_e = n_p = (1−x)n_H and η = α_A n_H/Γ. The positive physical root x = 2η/[1+2η+√(1+4η)] gives x_A ≈ 0.000998005 for η_A = 0.001 and x_B ≈ 0.00884156 for η_B = 0.009. This is assumed ionization equilibrium, not a solved relaxation, Saha, heating/cooling or pressure balance. Panel A plots each full number-column inventory on the same linear axis from zero to one in units N_ref. Blue electron columns equal the proton columns: about 0.899101796 for A and 0.892042600 for B. Orange neutral-H portions are about 0.000898204 and 0.007957400, summing with each blue proton count to total 0.9. The blue+orange stack is a hydrogen-nucleus inventory; an electron is counted with its corresponding ionized proton, not as extra hydrogen mass. A’s orange segment is honestly subpixel-sized on the full scale. Panel B therefore plots neutral columns alone on a separate linear zero-based axis from zero to 0.01 in the same N_ref units, exactly 100 times the pixels-per-column scale. It does not enlarge their physical shares in panel A or change the denominator. The labelled orange lengths give A 0.000898204 and B 0.007957400. B contains half the total hydrogen column yet contributes about 89.8572 percent of neutral column. Combined total hydrogen is 1.8N_ref, neutral 0.0088556045N_ref and free electrons 1.7911444N_ref; the hydrogen-column-weighted neutral fraction is about 0.00491978. Number columns have m⁻² units, derived from physical m⁻³ density times proper path length. No absolute N_ref, temperature, coefficient, density or path scale is assigned; it can be chosen optically thin. The common coefficient need not impose equal pressure in unequal-density slabs. This is not a line-strength observation, total intergalactic census, helium/metals model, time history or measured cosmic baryon fraction. Numerical captions under each full inventory bar explicitly label the ionized-H column on the left and neutral-H column on the right, so these categories do not rely on color alone. Original Ark teaching model.

A matter census needs complementary information

The common recombination coefficient can be consistent with a common prescribed temperature, but the toy solves no heating or cooling. Its unequal-density slabs need not be in pressure balance, and their future evolution is unspecified.

Real intergalactic ionization and spectra depend on radiation intensity and spectrum, density structure, temperature, velocities and thermal history. Helium, metals and collisional processes can also matter outside this pure-hydrogen construction.

An absorption component is not automatically a separate uniform cloud. Blending and the mapping from gas position and velocity into wavelength must be modelled before a line feature is turned into a physical inventory.

Combining tracers can help account for different gas states, but each conversion must state its assumptions and selection. The next lesson isolates optical depth, transmitted flux and instrument response in an explicitly synthetic absorption spectrum.

Worked reasoning

Compare the two neutral columns

Why does slab B dominate neutral absorption inventory even though both slabs have equal total hydrogen column?

Each total column is density times proper length: n_ref(0.9L) = (9n_ref)(0.1L) = 0.9N_ref.
The balance parameter is nine times larger in B, giving neutral fractions about 0.000998005 and 0.00884156.
Their neutral columns are about 0.000898204N_ref and 0.007957400N_ref, so B contributes about 0.898572 of their sum.

Density changes the conditional ionization fraction, so equal total columns do not imply equal neutral columns under the supplied radiation field.

Try the reasoning

May the neutral-hydrogen column be used directly as total hydrogen column in these mostly ionized slabs?
Show the answer and reasoning
No.
The ionization correction depends on the stated density, radiation and recombination balance.
Does assumed ionization equilibrium prove that heating, cooling and pressure are in equilibrium?
Show the answer and reasoning
No.
Those are separate physical balances, and their evolution is not solved here.
Does the free-electron column equal the total hydrogen-nucleus column in this model?
Show the answer and reasoning
No.
It equals the ionized proton column in pure hydrogen; the neutral nuclei are excluded from the free-electron count.

What to carry forward

Neutral absorption can trace only a small part of a largely ionized hydrogen inventory.
Equal total columns can have different neutral columns because ionization depends on local conditions.
Gas phases, thermal history, line mapping and tracer selection belong in any diffuse-matter census.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.

The Physics of the Intergalactic MediumAvery A. MeiksinTeaching resourcearxiv.orgRead source