The hydrogen 21-centimetre signal

Spin temperature describes a level-population ratio

Neutral hydrogen has hyperfine levels whose transition produces radiation near the wavelength conventionally called 21 centimetres. The line can probe neutral gas through its contrast against a background radiation field.

Define spin temperature T_s by n_1/n_0 = 3exp(−T_*/T_s), where n_1 and n_0 are the upper and lower hyperfine populations and T_* = hν_0/k_B is the splitting expressed as a temperature. The factor three is the ratio of their statistical weights.

This definition describes the hyperfine population ratio. It does not set T_s equal to the gas kinetic temperature. Radiation, collisions and other coupling processes determine how those populations relate to the gas and radiation fields.

The supplied example uses positive spin temperatures well above T_* and a Rayleigh–Jeans brightness account. It does not solve the population coupling, heating or ionization history that would establish T_s in a real region.

Line emission and background attenuation contribute together

For a uniform-excitation slab with optical depth τ and background rest-frame brightness temperature T_R, use T_out,rest = T_s(1 − exp(−τ)) + T_R exp(−τ) in the stated Rayleigh–Jeans limit.

The first term is the slab’s contribution to the emergent brightness, while the second is the attenuated background. Subtracting the unattenuated background gives T_out,rest − T_R = (T_s − T_R)(1 − exp(−τ)).

Under cosmological redshift, the Rayleigh–Jeans brightness-temperature difference observed at the redshifted line frequency is ΔT_obs = (T_s − T_R)(1 − exp(−τ))/(1 + z). Both temperatures in the numerator are defined in the slab rest frame.

This is a difference relative to the redshifted background, not an absolute sky temperature. Its sign can be negative, zero or positive even though the supplied slab contains neutral hydrogen in every case.

Neutral gas can absorb, have zero contrast or emit

Choose T_R = 40 K, τ = 0.02 and z = 9. Compare separate prescribed uniform slabs with T_s = 10 K, 40 K and 80 K. The 40 K background is a teaching input, not an observed CMB temperature.

At T_s = 10 K, the slab is cooler in spin temperature than its background, giving ΔT_obs ≈ −59.4040 mK. At T_s = 80 K, the same transfer formula gives positive contrast of about 79.2053 mK.

When T_s = T_R = 40 K, emission and background attenuation exactly cancel in this line-contrast account. The output rest brightness remains 40 K and ΔT_obs is zero despite the nonzero supplied optical depth.

These rows are different prescribed slabs with a common τ. They do not follow one fixed-column parcel as it warms: maintaining τ while changing T_s can require different columns or line profiles. No such column evolution is calculated here.

A rest-frame transition is observed at a lower frequency

The observed frequency is ν_obs = ν_0/(1 + z). For transparent teaching arithmetic, supply the rounded nominal rest frequency ν_0 = 1420 MHz. This is not a claim that the atomic rest frequency is exactly that rounded number.

With that input, redshifts z = 4, 9 and 19 correspond to frequencies 284, 142 and 71 MHz. The frequency relation labels the propagation redshift; it does not by itself determine spin temperature or neutral abundance.

A real spectral observation also involves instrumental response, foreground radiation and the relation between frequency and the gas sampled along the sightline. Velocity gradients and spatial structure can enter the line-opacity account.

The three brightness rows use a single supplied redshift and separate slab conditions. They are not a measured global spectrum or a predicted signal as a function of frequency across an inferred cosmic history.

Contrast is sensitive to more than the presence of neutral atoms

The sign in this example is controlled by T_s − T_R, while the magnitude also depends on optical depth and redshift. Neutral gas alone is therefore insufficient to specify a positive signal or a fixed brightness amplitude.

In a physical history, neutral density, hyperfine coupling, thermal evolution, velocity structure and ionization morphology can all affect the radiation reaching the observer. Their spatial variation carries information beyond one uniform slab.

Zero contrast does not uniquely mean no neutral gas: the T_s = T_R row supplies an explicit alternative. Inferring gas properties requires a model that distinguishes these possibilities and incorporates the observation uncertainties.

This calculation introduces transfer and sign conventions without adopting any present detection, upper limit or measured cosmic-temperature claim. Keep those inference outcomes separate from the recomputable behavior of the supplied slabs.

Worked reasoning

Try the reasoning

1. Does neutral hydrogen always produce positive 21-centimetre contrast?

   Show the answer and reasoning

   No.

   The supplied 10 K spin-temperature slab absorbs against its 40 K background, while the 80 K slab gives positive contrast.

2. Is spin temperature equal to gas kinetic temperature by definition?

   Show the answer and reasoning

   No.

   Spin temperature specifies hyperfine populations; the relevant coupling processes must establish any equality.

3. What frequency follows at z = 19 from the deliberately rounded 1420 MHz input?

   Show the answer and reasoning

   71 MHz.

   Divide the supplied rest frequency by 1 + z = 20, retaining its rounded teaching-input status.

What to carry forward

• Spin temperature describes hyperfine populations and needs its own coupling account.
• Line contrast depends on spin temperature relative to the background and can have either sign or vanish.
• A supplied slab calculation is distinct from a measured or inferred cosmological 21-centimetre history.