Cosmology · The thermal history of the early Universe

Equilibrium, decoupling and freeze-out

Follow a smooth kinetic-temperature departure from a cooling bath while the same species retains a fixed comoving particle number.

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 kinetic relaxation from chemical number-changing processes.
Interpret a supplied smooth temperature evolution through a rate crossing.
Explain why a cooler particle population does not automatically imply a smaller comoving abundance.

Equilibrium is a statement about specified properties

Kinetic equilibrium concerns momentum sharing and the form of a momentum distribution. Chemical equilibrium concerns reactions that adjust particle abundances consistently with conserved charges and bath conditions. The relevant processes and timescales can differ.

A species can stop changing its comoving number while it still exchanges momentum with a bath. Other model histories can have different orderings. There is no universal rule that chemical freeze-out must always precede kinetic decoupling.

Freeze-out describes a dynamical departure whose definition should name the quantity involved. A single temperature or rate equality is usually a useful approximate marker, rather than a discontinuous switch that turns all interactions off.

The supplied example tracks a nonrelativistic momentum second moment and a separately conserved particle number. It does not calculate a real dark-matter relic abundance, identify a detected particle or require the whole later distribution to remain exactly thermal.

A passive species exchanges heat but not number

Let q = a/a_ref run from 1 to 8. A radiation bath with constant entropy degrees of freedom has T_b = T_ref/q. Choose a passive homogeneous nonrelativistic species χ with mc²/(k_B T_ref) = 100, negligible effect on the background and no number-changing reactions during this interval.

Define its kinetic temperature T_χ from the mean nonrelativistic kinetic energy, ⟨p²/(2m)⟩ = (3/2)k_B T_χ. This defines a momentum moment even when a complete thermal distribution has not been established.

Use the supplied positive relaxation law dT_χ/dt + 2H T_χ = Γ_rel(T_b − T_χ), with Γ_rel/H = 16q⁻⁴ and initial T_χ = T_b at q = 1. Γ_rel is an effective kinetic relaxation coefficient, not a raw collision count.

Without the relaxation term, this nonrelativistic moment cools as a⁻². The bath cools only as a⁻¹. Positive heat exchange therefore competes with the faster free cooling of χ as the species becomes less coupled.

A rate crossing does not create a temperature jump

Write N = ln q and u = T_χ/T_b. The stated law becomes du/dN = 16e^(−4N)(1 − u) − u, with u = 1 initially. This differential equation is an optional way to reproduce the supplied table; the first comparison can be read directly from its columns.

At q = 2, Γ_rel/H = 1, but T_χ/T_b is already about 0.727603. At q = 4 the ratio is about 0.419282. The finite-rate solution changes smoothly through the equality, without an imposed sudden jump.

Exact equality of the temperatures at the initial time does not ensure exact equality thereafter. Expansion continuously changes the distribution that the relaxation process would need to maintain. The finite relaxation coefficient permits a lag.

The numerical temperature values below are solutions of the supplied moment law, independently checked by time stepping and an integrating-factor integral. Their displayed precision describes this constructed equation, not the accuracy of a real particle model.

A supplied passive nonrelativistic species in a radiation bath; chemical number-changing reactions are absent throughout
q	Γ_rel/H	T_b/T_ref	T_χ/T_b	T_χ/T_ref	Comoving number/reference
1	16	1	1	1	1
1.5	3.160494	0.666667	0.847158	0.564772	1
2	1	0.5	0.727603	0.363802	1
3	0.197531	0.333333	0.539491	0.179830	1
4	0.0625	0.25	0.419282	0.104821	1
8	0.00390625	0.125	0.215647	0.0269559	1

Cooling does not remove particles from a comoving region

The assumed number law is n_χa³ = constant. The physical number density falls as q⁻³ because the chosen comoving region expands, while the number inside that region remains fixed.

For example, at q = 2 the physical number density is one eighth of its reference value, yet the comoving number ratio in the table remains 1. This geometric dilution is distinct from number-changing annihilation or creation.

At q = 8 the moment temperature is about 0.026956 T_ref, above the value 1/64 T_ref that a population freely cooling from the initial time with no later heat exchange would have. Some bath heating has occurred even though the late relaxation ratio is small.

Neither this temperature comparison nor the fixed comoving number is a derived freeze-out yield from a microscopic reaction network. The number conservation was supplied as a condition; a chemical abundance calculation would require its own rate equation and initial data.

The supplied model follows a homogeneous passive nonrelativistic species χ with mc²/(k_B T_ref)=100, negligible background backreaction, no number-changing reactions and 1≤q=a/a_ref≤8. The radiation bath has constant entropy degrees of freedom and T_b/T_ref=1/q. Its momentum-moment temperature obeys dT_χ/dt+2HT_χ=Γ_rel(T_b−T_χ), with defined effective Γ_rel/H=16q^(−4) and initial T_χ=T_b. The upper graph has linear q and logarithmic base-ten temperature-ratio axes. Dashed orange squares show the bath 1/q; solid blue circles show the smooth numerical moment solution; dotted violet diamonds show the separate no-heat-exchange-from-start reference 1/q². At q=1,1.5,2,3,4,8 the moment ratios T_χ/T_ref are 1, approximately 0.564772,0.363802,0.179830,0.104821,0.0269559. A vertical guide marks q=2, where Γ_rel/H=1 and T_χ/T_b is already approximately 0.727603, without a discontinuity. At q=8 the moment remains above the free-from-start value 1/64 because earlier heat exchange matters. The lower panel uses the same linear q mapping with a separate linear number-ratio axis; a dashed green square-marked track stays exactly at 1 under the supplied law n_χa³=constant. Physical number density still dilutes as q^(−3), so at q=2 it is 1/8 of reference while comoving number remains 1. Γ_rel is the coefficient defined by this moment equation, not a raw collision count. A kinetic moment does not guarantee an exactly thermal full distribution after decoupling. The calculated decimals are arithmetic for the chosen law, not measured particle precision, a fitted WIMP scenario, a microscopic cross-section, a chemical freeze-out yield or a universal ordering of kinetic and chemical decoupling. Original Ark teaching model.

A useful decoupling estimate still needs a kinetic check

The equality Γ_rel = H marks q = 2 in this chosen law. It helps locate a transition region, while the actual solution shows the gradual departure and memory of earlier heat exchange.

Different definitions of an effective momentum-transfer rate can carry different numerical factors. The rate must be paired with the evolution equation that defines it. An unrelated raw scattering rate should not be substituted without deriving the connection.

A full distribution treatment may be needed when a moment equation is insufficient or when kinetic and chemical processes affect each other. The passive moment example deliberately excludes those complications rather than claiming that they never matter.

State what froze, which interactions remain, which bath is used and what was assumed about the initial state. Those details prevent the phrase “out of equilibrium” from being used as if it described every property in the same way.

Worked reasoning

Check the number account at the rate crossing

At q = 2 the species is cooler than the bath. Does the supplied model predict that particles have disappeared from a comoving region?

The temperature solution gives T_χ/T_b ≈ 0.727603 at Γ_rel/H = 1.
The separate number law is n_χa³ = constant, so the comoving number ratio is still 1.
The physical density falls to 1/8 of its reference value through volume expansion, without a number-changing reaction.

No. Kinetic cooling and the comoving particle number are separate accounts in this model.

Try the reasoning

Does Γ_rel = H force T_χ to jump discontinuously?
Show the answer and reasoning
No.
The stated finite-rate differential equation has a smooth solution.
Does assigning T_χ from ⟨p²/(2m)⟩ prove the whole post-decoupling distribution is exactly thermal?
Show the answer and reasoning
No.
A momentum moment can be defined without establishing the full distribution.
Must chemical freeze-out occur before kinetic decoupling in every particle model?
Show the answer and reasoning
No.
The relevant processes can interact and their ordering depends on the specified model.

What to carry forward

Kinetic and chemical equilibrium concern different properties and rates.
The supplied kinetic-temperature solution crosses Γ_rel/H = 1 smoothly.
A changing temperature or physical density does not by itself establish a changing comoving particle number.

References

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