Cosmology · Baryons, gas and cosmic neutrinos

Finding diffuse baryons through different tracers

Hold a supplied hydrogen column fixed while changing density and path length, separating a linear column from quadratic local recombination emission.

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 hydrogen and electron columns from a recombination emission measure.
Calculate integrated emission when both density and path length change.
Explain why an ionization correction and phase selection enter a diffuse-matter inference.

A weak signal need not mean a small material column

Diffuse gas can be difficult to account for because different observations weight its density, ionization, temperature and composition differently. A selected absorption or emission feature is one part of a physical inference.

A column counts material along a path: N_H = ∫n_H ds for total hydrogen nuclei. A recombination-line emission measure instead contains products of ion and electron number density.

Gas spread over a longer path can keep the same column while changing its emissivity. The distinction matters before turning a faint line into an estimate of total baryonic material.

The comparison below supplies four alternative pure-hydrogen slabs at one epoch. Their temperature and charge state are prescribed; no observed line, detection threshold or measured cosmic missing fraction is used.

Change density and path together

Choose positive proper density n_* and proper length L_*. Define N_* = n_*L_* and EM_* = n_*²L_*. Their units are m⁻² and m⁻⁵ respectively.

Use ionized fraction f_ion = 0.99, so n_e = n_p = f_ion n_H and n_HI = (1−f_ion)n_H. Here f_ion explicitly denotes the ionized fraction, keeping it distinct from a neutral-fraction convention.

Choose density ratios d = n_H/n_* equal to 1, 1/2, 1/4 and 1/8, and set each path length to L_*/d. Every total hydrogen column is then N_H = N_*.

All slabs therefore have neutral column 0.01N_* and free-electron and proton columns 0.99N_*. These equal columns are inputs of the comparison, while the densities and path lengths differ.

An n² emissivity does not fix the integrated density scaling

For each uniform slab, EM = ∫n_en_p ds = f_ion²n_H²L. Substituting L = L_*/d gives EM/EM_* = f_ion²d = 0.9801d.

At fixed common temperature T_*, hold a chosen effective recombination-line coefficient α_eff(T_*) constant. In a local optically thin isotropic emission model, the intrinsic frequency-integrated line intensity is I = hν_lineα_eff EM/(4π).

The local emissivity falls by a factor of four when density halves at fixed charge fraction. The doubled path then restores a factor of two, so the integrated emission measure and intrinsic intensity fall by a factor of two.

The table’s faintest slab has the same total hydrogen column as the first but one-eighth its intrinsic line intensity. No cosmological dimming, extinction, instrumental response or line transfer is added to this local comparison.

Alternative prescribed pure-H slabs with f_ion = 0.99 and common temperature/coefficient; no equilibrium or observational fit
n_H/n_*	L/L_*	N_H/N_*	N_HI/N_*	N_e/N_*	EM/EM_*	I/I_first
1	1	1	0.01	0.99	0.9801	1
1/2	2	1	0.01	0.99	0.49005	1/2
1/4	4	1	0.01	0.99	0.245025	1/4
1/8	8	1	0.01	0.99	0.1225125	1/8

A fixed neutral column can imply different total columns

Consider a separate comparison holding only N_HI = 0.01N_* fixed. For a uniform prescribed fraction, N_H = N_HI/(1−f_ion).

If f_ion = 0.90, the total is 0.1N_*. If f_ion = 0.99, it is N_*. The same supplied neutral column is compatible with a tenfold difference in total hydrogen under these different assumptions.

This second comparison does not change the first four slabs in time. It exposes a separate missing-input ambiguity: a neutral tracer alone does not supply its own ionization correction.

No ionization equilibrium has been solved to select either fraction. Radiation, density and temperature would need to enter a physical charge-state model before that correction could be inferred for actual gas.

Four alternative uniform pure-hydrogen slabs share one epoch, prescribed common temperature/coefficient and ionized fraction f_ion =0.99. This is the ionized fraction, not the neutral fraction. Proper reference density n_* and path L_* define N_* =n_*L_* in m⁻² and EM_* =n_*²L_* in m⁻⁵. Choose density ratios d =1, 1/2, 1/4, 1/8 and paths L/L_* =1, 2, 4, 8. Panel A uses a common linear zero-based path scale from zero to 8 L_*; labels identify the four separate density states. Every total hydrogen column is exactly N_*. The neutral column is 0.01N_* and electron/proton columns each 0.99N_* in all four states. Panel B uses linear density d from 1/8 to 1 and a normalized ordinate zero to one. Dashed blue circles show constant N_H/N_* =one. Solid orange squares show EM/EM_* =f_ion²d =0.9801d, with samples 0.1225125, 0.245025, 0.49005, 0.9801 in increasing density order. Local recombination emissivity is quadratic in density, but integrating the doubled proper path when density halves restores a factor two; the integrated signal halves rather than quarters. At fixed symbolic effective recombination coefficient, intrinsic frequency-integrated line intensity is I =hν_line α_eff EM/(4π), in W m⁻² sr⁻¹. Thus I/I_first =d and the lowest-density state has one-eighth of the first intensity despite equal material column. This is a local optically thin isotropic emission comparison; it includes no cosmological dimming, extinction, measured line, noise, instrumental threshold or line transfer. The footnote is a separate fixed-neutral-column comparison: for prescribed N_HI =0.01N_*, f_ion =0.90 gives total 0.1N_*, whereas f_ion =0.99 gives total 1N_*. These alternatives differ tenfold and do not evolve the first four slabs. No ionization equilibrium is solved to choose either fraction; radiation, density and temperature are missing physical inputs. Fixed charge state and common coefficient do not solve thermal or mechanical equilibrium. No helium/metals, missing-baryon percentage, current census or observational mass conversion is supplied. Original Ark teaching model.

A census must carry phase and selection information

The line-intensity units provide a check: hν has joules, α_eff has m³/s and EM has m⁻⁵, giving energy per time per area per solid angle after the isotropic factor.

Real multiphase gas can involve helium, metals, temperature-dependent ion fractions, nonuniform density and radiation. Metal-ion signals additionally require an abundance conversion, while saturation and selection affect which systems are measured.

Tracer-selected categories can overlap, and a common piece of gas should be counted once in a declared mass account. Complementary tracers help only when their conversion assumptions and shared uncertainties are retained.

A useful diffuse-baryon inference states the measured tracer, physical path and geometry, charge-state and thermal model, abundance assumptions and observing selection. The two slab examples show why the signal-to-material conversion belongs in the result.

Worked reasoning

Track both factors in the integrated emission

When density halves and the path doubles at fixed total column and f_ion, does the supplied line intensity fall to one quarter?

The local n_en_p product scales as density squared, so it becomes one quarter.
The path length is twice as large.
Their product gives half the emission measure and, with the same line coefficient, half the intrinsic line intensity.

It halves in this specified comparison. The local quadratic density law must be integrated over the changing path.

Try the reasoning

Does a weak selected recombination line by itself establish a small total hydrogen column?
Show the answer and reasoning
No.
A longer lower-density path can have the same column but weaker emission under the supplied fixed-charge conditions.
For N_HI = 0.01N_* and f_ion = 0.90, what total hydrogen column is required?
Show the answer and reasoning
0.1N_*.
Divide the neutral column by the neutral fraction 0.10.
Do the prescribed fractions and slab table prove a time-dependent ionization or thermal equilibrium solution?
Show the answer and reasoning
No.
They are alternative declared states; the processes selecting or maintaining them are outside the model.

What to carry forward

A column and a density-squared emission measure weight gas differently.
Changing path length can change the integrated scaling even when local emissivity is quadratic.
Ionization, composition, phase and selection assumptions must accompany a diffuse-matter estimate.

References

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