Cosmology · Cosmic dark ages, first light and reionisation

Ionising radiation and intergalactic gas

Derive a local pure-hydrogen ionization and recombination balance, keeping photon emissivity, per-atom rate and volume filling fraction distinct.

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

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After reading, you can

Distinguish an escaping photon budget from a local per-atom photoionization rate.
Derive the density dependence of recombination in a supplied pure-hydrogen cell.
Read a conditional approach to ionization equilibrium without treating it as a reconstructed cosmic history.

Emitted photons and local interaction rates are different quantities

An escaped ionizing-photon budget describes radiation leaving sources. The photoionization rate at a location also depends on how that radiation propagates, its spectrum and the cross section for the relevant neutral atom.

For hydrogen photoionization, photons above the ionization threshold contribute according to their incident photon flux and the frequency-dependent cross section. A comoving emissivity alone does not specify that local flux or its attenuation.

Write Γ for a supplied photoionization rate per neutral hydrogen atom, with units of inverse time. Multiplying Γ by the neutral-atom density gives ionizations per unit physical volume per unit time.

Recombination involves encounters of free electrons and protons. Its corresponding per-volume rate contains a recombination coefficient times n_e n_p. The coefficient has volume-per-time units and is not the same kind of quantity as Γ.

A fixed-density cell makes the bookkeeping explicit

Consider a prescribed homogeneous local pure-hydrogen cell with fixed proper total hydrogen density n_H, fixed temperature and a constant external Γ. Let x = n_p/n_H = n_e/n_H, so neutral density is n_H(1 − x).

Choose a case-B, on-the-spot recombination closure: photons from captures directly to the ground state are assumed to cause local ionizations and are excluded from the net recombination sink. The effective coefficient α_B is held positive and constant, and Γ counts the supplied external field.

Under these conditions, photoionizations add Γn_H(1 − x) per volume and recombinations remove α_B n_H²x². Dividing by the fixed total n_H gives dx/dt = Γ(1 − x) − α_B n_Hx².

This is a conditional local kinetic account. Cosmic dilution, gas dynamics, helium, collisional ionization, temperature evolution and radiation-transfer feedback are omitted. Case B is an assumption of this construction, not a universal approximation for all intergalactic gas.

An initially neutral cell need not become completely ionized

Define dimensionless u = α_B n_Ht and g = Γ/(α_B n_H). The equation becomes dx/du = g(1 − x) − x². Choose g = 1, x(0) = 0 and display only 0 ≤ u ≤ 4.

The initially neutral state begins ionizing, but the gain falls as fewer neutral atoms remain. The recombination sink increases with x² because both the electron and proton densities increase with x.

At u = 1, the supplied solution has x ≈ 0.530330. By u = 4, x ≈ 0.617923. These dimensionless points approach a balance below one; they are not assigned physical times or a cosmic reionization redshift.

The total number of hydrogen nuclei in this fixed cell remains unchanged. The table tracks charge state, not creation or destruction of nuclei, and the external radiation field is an imposed input rather than a self-consistent source-and-transfer calculation.

Supplied fixed proper-density pure-hydrogen stage with constant g = 1 and case-B closure; u = α_B n_Ht
u	Ionized atomic fraction x	Neutral atomic fraction 1 − x
0	0	1
0.5	0.369806	0.630194
1	0.530330	0.469670
2	0.608320	0.391680
4	0.617923	0.382077

Continuing recombination leaves a residual neutral population

At equilibrium, g(1 − x_eq) = x_eq². For g = 1 the physical root is x_eq = (√5 − 1)/2 ≈ 0.618034. The negative root is outside the allowed atomic-fraction interval.

A larger supplied g changes the balance: g = 0.25 gives x_eq ≈ 0.390388, while g = 4 gives x_eq ≈ 0.828427. These are comparisons among different imposed external-rate-to-recombination ratios, not measured gas states.

The equilibrium is stable within this scalar equation because the derivative of its right-hand side is −g − 2x, negative for positive g and physical x. Small departures from the equilibrium are driven back toward the balance under the fixed conditions.

The same numerical quadratic root can appear in other teaching examples with different physics. Here it follows from photoionization and recombination rates; it is not a Saha chemical-equilibrium calculation or evidence that the two descriptions have identical domains.

A prescribed homogeneous pure-hydrogen cell has fixed proper total hydrogen density n_H, fixed temperature and constant externally supplied photoionization rate Γ per neutral atom. The local atomic ionized fraction is x = n_p/n_H = n_e/n_H and neutral density is (1−x)n_H. A positive constant case-B coefficient α_B has volume-per-time units; the on-the-spot closure assumes captures directly to the ground state recycle their photons locally, excluding them from the net sink. The fixed-cell equation is dx/dt = Γ(1−x) − α_B n_H x². Use dimensionless u = α_B n_H t and g = Γ/(α_B n_H) = 1, initially x = 0, over 0 to 4. The graph uses a linear fraction axis from zero to one. Solid blue circles show x and dashed orange squares show 1−x. At u = 0, 0.5, 1, 2, 4, ionized fractions are 0, 0.369806, 0.530330, 0.608320, 0.617923, with their complementary neutral fractions. The green dotted guide is the stable equilibrium x_eq = (√5−1)/2, approximately 0.618034. The trajectory approaches it below unity because gain g(1−x) balances recombinations x². A separate table gives equilibrium x for g = 0.25, 1, 4 as 0.390388, 0.618034, 0.828427; these are different imposed rate ratios, not observed gas states. Hydrogen nuclei are conserved in the fixed cell. No physical time, measured coefficient, cosmic dilution, helium, gas dynamics, thermal evolution, collisional ionization or transfer feedback is calculated. Case B is conditional, not universal intergalactic closure. This local atomic fraction is not an HII bubble volume filling factor, and its rate balance is not Saha chemical equilibrium or a reconstructed reionization history. Original Ark teaching model.

A cosmic history needs spatial and radiative information

The local atomic fraction x counts ionized hydrogen nuclei relative to total nuclei in this prescribed homogeneous cell. A volume filling factor for ionized regions instead measures the fraction of a larger volume occupied by a defined phase.

Those quantities need not coincide in a patchy medium. Density variations, partially ionized gas, source locations, recombinations and radiative transfer affect how local fractions combine into a spatial ionization history.

Consequently, the local x² sink derived here should not be inserted unchanged into an equation for the volume filling factor of a two-phase bubble model. Each average requires its own definitions and closure assumptions.

A useful chain of questions is to identify the source emissivity, propagate the radiation, compute local interaction rates and then specify the spatial average being compared with observations. The finite cell example isolates the local-rate step in that chain.

Worked reasoning

Find the equilibrium of the supplied cell

Does constant g = 1 drive x to exactly one in the stated rate equation?

Set dx/du = 0, giving 1 − x_eq − x_eq² = 0.
Solve the quadratic and retain the root between zero and one: x_eq = (√5 − 1)/2.
The result is about 0.618034; photoionization of the remaining neutral atoms balances recombination of free electrons and protons.

No. Continuing recombination gives the sub-unity equilibrium x_eq ≈ 0.618034 under these fixed conditions.

Try the reasoning

Do α_B and Γ have the same units?
Show the answer and reasoning
No.
α_B has volume-per-time units, while Γ is an inverse-time rate per neutral atom; α_B n_H is an inverse-time combination.
Can a comoving escaped-photon emissivity be used directly as Γ?
Show the answer and reasoning
No.
Propagation, photon flux, spectrum and cross-section weighting are needed to determine the local per-atom rate.
Is x automatically the volume filling factor of ionized bubbles?
Show the answer and reasoning
No.
It is a local atomic fraction in a homogeneous cell; a spatial phase-volume average has a different definition and closure.

What to carry forward

A local photoionization rate needs radiation transport and cross-section weighting beyond emitted photon counts.
Recombination depends on the product of electron and proton densities.
Local atomic fractions and global ionized-volume filling factors must retain their separate meanings.

References

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