Dark ages after last scattering

Existing photons precede the first stars

The cosmic dark ages refer to the interval after the main recombination and last-scattering era and before the first stars. The name distinguishes the absence of those early luminous sources from the later era of first light.

It does not mean that the Universe contains no photons. The thermal background already exists, and much of that radiation propagates with reduced scattering as the free-electron abundance falls. Last scattering is not the creation of the entire background radiation field.

Mostly neutral intergalactic gas and existing radiation can therefore coexist. A description of the dark ages should also avoid claiming that every atom is neutral or that all interactions have ceased; the relevant residual populations and processes belong in a more detailed history.

Later luminous sources add radiation that can heat and ionize gas. Their formation and the subsequent reionization account are separate from the earlier neutral-atom formation and declining scattering opacity.

Radiation and gas need not keep the same temperature

An initially thermal freely redshifting radiation field has temperature T_R proportional to a⁻¹ when absorption, injection and entropy transfer are excluded. This relation describes the radiation distribution under those conditions.

For a separate homogeneous nonrelativistic ideal monatomic gas with fixed particle number and composition, adiabatic expansion without heating gives T_g proportional to n_g^(2/3). With n_g proportional to a⁻³, the gas kinetic temperature falls as a⁻².

The gas law assumes that the stated ideal-gas thermodynamic account applies. It is not automatically the law of a gas receiving Compton heating, stellar or X-ray energy, molecular cooling, compression in a forming structure or changing chemical contents.

A gas kinetic temperature and a radiation temperature describe different populations. Equality at one reference state does not require continuing equality after their relevant energy exchange becomes ineffective.

A chosen shared reference separates into two curves

Choose q = a/a_ref over 1 ≤ q ≤ 8 and supply T_ref = 100 K for both populations at q = 1. Apply T_R = 100 K/q, T_g = 100 K/q² and n_g/n_ref = q⁻³ under the preceding exclusions.

At q = 2, radiation has temperature 50 K while gas has temperature 25 K. At q = 4, the corresponding temperatures are 25 K and 6.25 K. The gas-to-radiation temperature ratio is 1/q.

The table uses a chosen reference temperature and dimensionless scale-factor interval. It does not locate a physical thermal-decoupling date, a redshift of first-star formation or the actual gas temperature at a measured epoch.

Both populations cool, but at different rates in this construction. Their divergence is a consequence of the separate energy accounts, rather than a disappearance of background radiation.

The real history requires more than a free-cooling law

Energy transfer through residual electrons can influence gas temperature when its relevant rate is effective. Establishing when it becomes negligible requires a rate and expansion calculation. The chosen q = 1 reference does not perform that calculation.

As structure develops, gas can compress and exchange energy through processes excluded from the homogeneous toy. First-source formation needs gravity, gas dynamics, chemistry and cooling, rather than a uniform temperature curve alone.

Once luminous sources contribute heating and ionizing radiation, the no-heating assumption must be reconsidered. Extending T_g ∝ a⁻² unchanged through a photoheated stage would contradict the example’s own conditions.

The hydrogen hyperfine populations used in 21-centimetre studies introduce a spin temperature with its own coupling processes. It should not be set equal to this gas kinetic temperature merely because both quantities are called temperatures.

Neutralization, first light and reionization ask different questions

Recombination studies ask how electrons and nuclei become bound and how scattering changes. Dark-age studies follow matter, radiation and the developing structures before the first luminous sources. First-light studies ask how those sources form and radiate.

Reionization then concerns how later radiation changes the ionization state of intergalactic gas, together with its spatial distribution and thermal response. An ionized fraction, a source population and an observation of absorption describe different parts of that account.

The simple cooling calculation is useful because its assumptions are explicit and its temperatures can be recomputed. It provides a limiting comparison against which additional heating or coupling terms can be understood.

To move from that limit to a history, specify the populations, interactions, source evolution and observations. Keep the distinction between a supplied background example and an inferred cosmic chronology.

Worked reasoning

Try the reasoning

1. Do the cosmic dark ages mean that no background photons exist?

   Show the answer and reasoning

   No.

   Existing thermal background radiation remains; the name refers to the interval before the first stars.

2. Can the same unheated gas law be extended unchanged through a later photoheated stage?

   Show the answer and reasoning

   No.

   An additional energy source violates the no-heating condition and needs a new thermal-evolution account.

3. Does the table establish the 21-centimetre spin temperature?

   Show the answer and reasoning

   No.

   It supplies a gas kinetic temperature; hyperfine populations and their couplings require separate reasoning.

What to carry forward

• Dark ages do not mean a Universe without existing background radiation.
• Freely cooling ideal gas and radiation have different temperature laws under stated conditions.
• First light and reionization add source, heating and ionization processes beyond the homogeneous cooling limit.